Class 12 Physics Chapter 6: Orthographic Projection of Right Regular Solids NCERT Solutions
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Complete Chapter Solutions
Project the front view and top view of a square prism of 35 mm base edges and 50 mm vertical height, resting on H.P., with two vertical rectangular faces parallel to V.P.
Concept used. When the axis is perpendicular to H.P., the base is seen in true shape in the top view. Start with the top view and project the front view.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the square prism using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
View-selection check
The answer is correct when the top view shows the base in true shape and the front view shows the full vertical height.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the square prism using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
A triangular prism of 40 mm base edges and 60 mm height stands on H.P. with one vertical rectangular face at the rear, parallel to V.P. Draw its projections.
Concept used. When the axis is perpendicular to H.P., the base is seen in true shape in the top view. Start with the top view and project the front view.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the triangular prism using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
View-selection check
The answer is correct when the top view shows the base in true shape and the front view shows the full vertical height.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the triangular prism using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
Draw the projections of a cylinder resting on H.P. on its base, with 30 mm base diameter and 40 mm long axis.
Concept used. When the axis is perpendicular to H.P., the base is seen in true shape in the top view. Start with the top view and project the front view.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the cylinder using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
View-selection check
The answer is correct when the top view shows the base in true shape and the front view shows the full vertical height.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the cylinder using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
Project the front view and top view of a hemisphere resting on H.P. with its circular face on top. Diameter is 60 mm.
Concept used. When the axis is perpendicular to H.P., the base is seen in true shape in the top view. Start with the top view and project the front view.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the hemisphere using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
View-selection check
The answer is correct when the top view shows the base in true shape and the front view shows the full vertical height.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the hemisphere using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
Project the front view and top view of the frustum of a hexagonal pyramid, 25 mm base edges and 70 mm height, cut at mid-height parallel to its base.
Concept used. When the axis is perpendicular to H.P., the base is seen in true shape in the top view. Start with the top view and project the front view.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the hexagonal pyramid frustum using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
View-selection check
The answer is correct when the top view shows the base in true shape and the front view shows the full vertical height.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw the \(XY\) reference line and keep enough space for both views.
- Below \(XY\), draw the true-shape top view of the hexagonal pyramid frustum using the given base size.
- Name all base corners or rim points in sequence.
- Project vertical construction lines upward from every named point.
- Set the given axis length or height in the front view.
- Join visible edges with thick object lines and show hidden rear edges with dashed lines only when required.
- Check that the top view carries the base size and the front view carries the height.
Project the front view and top view of a hollow cylinder having outer diameter 50 mm, inner diameter 40 mm and length 50 mm, resting on H.P. with its axis perpendicular to V.P.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the hollow cylinder above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the hollow cylinder above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A triangular pyramid of 50 mm base and 50 mm axis rests on its base corner on H.P., so that the upper edge of the base is horizontal. The base is at the rear and parallel to V.P. Draw projections.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the triangular pyramid above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the triangular pyramid above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
Project the front view and top view of a pentagonal prism of 30 mm base edges and 60 mm long edges perpendicular to V.P., with its rectangular face on top parallel to H.P.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the pentagonal prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the pentagonal prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A frustum of a square pyramid of 40 mm base edges and 20 mm top edges rests on H.P. on a base edge. Its 50 mm axis is horizontal and at right angles to V.P.; the cut face is in front. Draw projections.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the square pyramid frustum above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the square pyramid frustum above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A hexagonal prism of 25 mm base edge and 60 mm axis rests on one base edge on H.P. and its axis is perpendicular to V.P. Project its front view and top view.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the hexagonal prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the hexagonal prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
Project the front view and top view of a cylinder with base diameter 50 mm and height 70 mm, resting on H.P., with its axis perpendicular to V.P.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the cylinder above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the cylinder above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
Draw the front view and top view of a cone of base diameter 30 mm and axis 65 mm, with its axis perpendicular to V.P., keeping the vertex in front.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the cone above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the cone above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A square prism, base 40 mm side and axis 70 mm long, lies on one rectangular face. Its axis is perpendicular to V.P. Draw its front view and top view.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the square prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the square prism above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
The frustum of a triangular pyramid of 50 mm base edge and 20 mm top edge rests on H.P. with its base edge on it. The 60 mm axis is parallel to H.P. and at right angles to V.P. The cut face is in front. Project its views.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the triangular pyramid frustum above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the triangular pyramid frustum above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A right regular pentagonal pyramid of base edge 25 mm and height 60 mm has its axis perpendicular to V.P. and its base parallel to V.P. Draw its projections.
Concept used. When the axis is perpendicular to V.P., the end face is seen in true shape in the front view. Start with the front view and project the top view.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the pentagonal pyramid above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
View-selection check
The answer is correct when the front view shows the end face in true shape and the top view shows the full axis length.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and mark the front-view space above it.
- Draw the true-shape front view of the pentagonal pyramid above \(XY\).
- Name all corner, rim or generator points clearly.
- Project downward from each point to form the top view.
- Set the given axis length in top view, keeping it perpendicular to V.P.
- Join corresponding points, darken visible outlines and add dashed hidden edges where the rear face is not visible.
- Check that the front view carries the end-face shape and the top view carries the length.
A hexagonal prism, base 25 mm side and axis 60 mm long, lies on the ground on one face with the axis parallel to both V.P. and H.P. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., the side or profile view helps fix the end shape, while the front and top views show the axis parallel to \(XY\).
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the hexagonal prism end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
View-selection check
The answer is correct when the axis is parallel to \(XY\) in both views and the given resting condition is still visible.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the hexagonal prism end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
A triangular pyramid, base 25 mm side and axis 50 mm long, rests on the ground on one base edge. Its axis is parallel to both planes. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., the side or profile view helps fix the end shape, while the front and top views show the axis parallel to \(XY\).
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the triangular pyramid end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
View-selection check
The answer is correct when the axis is parallel to \(XY\) in both views and the given resting condition is still visible.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the triangular pyramid end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
A cylinder, base 40 mm diameter and axis 60 mm long, lies on the ground on its generators with the axis parallel to both V.P. and H.P. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., the side or profile view helps fix the end shape, while the front and top views show the axis parallel to \(XY\).
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the cylinder end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
View-selection check
The answer is correct when the axis is parallel to \(XY\) in both views and the given resting condition is still visible.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the cylinder end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
A frustum of a hexagonal pyramid of 20 mm base edges and 10 mm top edges rests on H.P. on a base edge with its 50 mm axis horizontal and parallel to V.P. The cut top face is in front. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., the side or profile view helps fix the end shape, while the front and top views show the axis parallel to \(XY\).
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the hexagonal pyramid frustum end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
View-selection check
The answer is correct when the axis is parallel to \(XY\) in both views and the given resting condition is still visible.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Draw \(XY\) and first set the contact condition with H.P.
- Draw the view in which the hexagonal pyramid frustum end shape or resting face is easiest to control.
- Keep the axis horizontal, so it remains parallel to \(XY\).
- Project the other view using perpendicular projectors.
- Transfer the axis length without changing it.
- Mark visible boundaries with object lines and hidden boundaries with dashed lines.
- Verify that the solid rests on the stated face, edge or generator.
A triangular prism with 25 mm edges at its base and axis 60 mm long rests on one base edge with axis parallel to V.P. and inclined at \(30^\circ\) to H.P. Draw projections.
Concept used. When the axis is parallel to V.P. and inclined to H.P., solve it in two steps. First draw the simple position with axis perpendicular to H.P., then tilt the front view.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
View-selection check
The answer is correct when the final front-view axis has the given H.P. inclination and the top view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
A pentagonal pyramid of 25 mm base edges and 50 mm axis has its axis perpendicular to V.P. and 50 mm above H.P. Draw projections if one base edge is inclined at \(30^\circ\) to H.P.
Concept used. When the axis is parallel to V.P. and inclined to H.P., solve it in two steps. First draw the simple position with axis perpendicular to H.P., then tilt the front view.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
View-selection check
The answer is correct when the final front-view axis has the given H.P. inclination and the top view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
A frustum of square pyramid of 20 mm top edges, 40 mm bottom edges and 50 mm axis has its side face inclined at \(45^\circ\) to H.P. with axis parallel to V.P. Draw projections.
Concept used. When the axis is parallel to V.P. and inclined to H.P., solve it in two steps. First draw the simple position with axis perpendicular to H.P., then tilt the front view.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
View-selection check
The answer is correct when the final front-view axis has the given H.P. inclination and the top view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
The frustum of a cone has 90 mm base diameter and 30 mm top diameter. Draw projections when its axis is parallel to V.P. and inclined at \(30^\circ\) to H.P.
Concept used. When the axis is parallel to V.P. and inclined to H.P., solve it in two steps. First draw the simple position with axis perpendicular to H.P., then tilt the front view.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
View-selection check
The answer is correct when the final front-view axis has the given H.P. inclination and the top view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
The frustum of a cone has 90 mm base diameter and 30 mm top diameter. Draw projections when its axis is parallel to V.P. and inclined at \(60^\circ\) to H.P.
Concept used. When the axis is parallel to V.P. and inclined to H.P., solve it in two steps. First draw the simple position with axis perpendicular to H.P., then tilt the front view.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
View-selection check
The answer is correct when the final front-view axis has the given H.P. inclination and the top view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to H.P. and draw the simple top view first.
- Project the first front view and show the true axis length.
- Redraw this front view so that the axis makes the required angle with \(XY\).
- Project downward from every point of the inclined front view.
- Draw horizontal loci from the first top view.
- Locate intersections between new projectors and old loci to form the final top view.
- Join matching points and decide visible and hidden edges from the final position.
Draw the projections of a pentagonal prism having 25 mm edge of its base and axis 50 mm long when it is resting on its base with an edge of its base inclined at \(30^\circ\) to V.P.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
A triangular pyramid of 50 mm base edges and 60 mm axis has one base corner touching V.P., with axis parallel to H.P. and inclined at \(45^\circ\) to V.P. Draw projections.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
A frustum of a cone of 40 mm base diameter and 20 mm cut-face diameter rests on H.P. Its 50 mm axis is parallel to H.P. and inclined to V.P. at \(30^\circ\) towards right. Project top view and front view.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
A square duct is a frustum of a square pyramid. The top and bottom sides are 90 mm and 60 mm, and the length is 110 mm. Its axis is parallel to H.P. and inclined at \(60^\circ\) to V.P. Draw projections, neglecting sheet thickness.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
Draw projections of a square prism having 30 mm base edge and 55 mm axis when it rests on its base with its axis inclined at \(30^\circ\) to V.P.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
Draw projections of a hexagonal prism having 20 mm base edge and 50 mm axis when it rests on its base, with axis parallel to H.P. and inclined at \(40^\circ\) to V.P.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
A triangular prism of 50 mm base edges and 60 mm axis rests on its base and has its axis inclined at \(45^\circ\) to V.P. Draw projections.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
A hexagonal pyramid of 25 mm base edges and 60 mm axis rests on its base and has its axis inclined to V.P. at \(30^\circ\). Draw its projections.
Concept used. When the axis is parallel to H.P. and inclined to V.P., solve it in two steps. First draw the simple position with axis perpendicular to V.P., then tilt the top view.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.
View-selection check
The answer is correct when the final top-view axis has the given V.P. inclination and the front view is projected from it.Construction route. First decide which view shows the real shape. For a prism, pyramid, cylinder, cone or frustum, the view normal to the axis gives the end face. The other view carries the axis length. Keep the first position light, because it is only a construction aid. Then darken the final view after the tilt or projection is complete.
- Assume the axis is perpendicular to V.P. and draw the true end shape in front view.
- Project the first top view and set the axis length.
- Redraw this top view so that the axis makes the required angle with \(XY\).
- Project upward from every point of the inclined top view.
- Draw horizontal loci from the first front view.
- Locate intersections between new projectors and old loci to form the final front view.
- Join matching points and use dashed lines for hidden rear edges.








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