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The volume of a combination of solids is an important concept in geometry and is often used in real-world applications. This concept involves finding the total volume of a shape that is made up of multiple solid shapes combined.
- In engineering, architecture, and construction volume determination is very important.
- A solid understanding of how to calculate the volume of a combination of solids and its practical applications can be obtained through this method.
- It can be calculated by combining multiple 2D shapes.
Read More: NCERT Solution for Class 10 Surface Area and Volume
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Key Terms: Solid, Shapes, Volume, Cube, Cylinder, Cone, Sphere.
What is a Solid Shape?
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A solid shape, also known as a three-dimensional (3D) shape, is a geometric figure that has length, width, and height or depth. It is a physical object that occupies space and has a definite shape and volume. Unlike 2D shapes, such as squares or circles, which are flat and have only two dimensions (length and width), solid shapes have three dimensions and can be seen from all sides.
- Solid shapes are commonly found in the physical world, such as buildings, furniture, and objects around us.
- They are also essential in fields such as engineering, architecture, and construction, where they are used to design and create structures and products.
- In mathematics, solid shapes are studied as part of geometry, which deals with the properties and relationships of 2D and 3D figures.
Read More: Surface Areas and Volumes
Types of Solid Shapes and Their Volumes
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A few 2D shapes and their volumes are described as follows–
- Cuboid:
- A cuboid is a rectangular-shaped solid with six faces, where the opposite faces are congruent rectangles.
- The length, width, and height of a cuboid can be different from each other, but the opposite faces are always parallel and congruent.
- Let L, B, and H be the length, breadth, and height of a cuboid, then the volume of a cuboid can be given by the formula:
Volume of a Cuboid = L× B× H
Cuboid
- Cube:
- A cube is a special type of cuboid with six congruent square faces.
- The length, width, and height of a cube are all equal, and the opposite faces are parallel and congruent.
- Let a be the side of a cube, then the volume of a cube can be given by the formula:
Volume of a Cube = a × a × a = a3
Cube
- Cylinder:
- A cylinder is solid with two parallel circular bases connected by a curved surface.
- The height (h) of a cylinder is the distance between the two bases, and the radius (r) is the distance from the center of the base to the edge.
Cylinder
The volume of a cylinder= πr2h
Where, r = radius of the base of the cylinder
h = height of the cylinder
- Cone:
- A cone is solid with a circular base and a curved surface that tapers towards a single point called the apex.
- The height of a cone is the distance between the base and the apex, and the radius is the distance from the center of the base to the edge.
Volume of Cone = 13r2h
Here, r = radius of the base of the cone, h = height of the cone, l = slant height of the cone
Cone
- Sphere:
- A sphere is a solid with a curved surface that is perfectly round.
- All points on the surface of a sphere are equidistant from its center.
The volume of a Sphere = 43r3
Sphere
Where, r = radius of the sphere.
Read More: Surface Areas and Volumes Revision Notes
Importance in Engineering and Construction
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Understanding the concept of the volume of a combination of solids is essential in engineering and construction. These professionals are often tasked with designing and building structures that are not made up of simple shapes but instead have complex shapes. Having a good understanding of how to calculate the volume of combined solids can help in determining the amount of material required to build such structures.
- For instance, architects and civil engineers use volume calculations to determine the amount of concrete, steel, and other materials needed to construct buildings, bridges, and other structures.
- Accurate calculations of volume ensure that the structures are built to withstand the forces they are subjected to and remain safe and stable over time.
- The concept of volume of combined solids is also relevant in the manufacturing industry, particularly in designing and building products with complex shapes.
- The automotive industry uses volume calculations to design car parts and engines, ensuring that they fit together seamlessly and operate effectively.
Read More: Surface Area of a Cylinder Formula
Things to Remember
- A solid shape, also known as a three-dimensional (3D) shape, is a geometric figure that has length, width, and height or depth.
- Solid shapes are studied as part of geometry, which deals with the properties and relationships of 2D and 3D figures.
- Cuboids, cubes, cylinders, cones, and spheres are examples of such 2D shapes.
- Building structures can be possible with the help of such 2D shapes.
Sample Questions
Ques: A water tank has a cylindrical base with a radius of 2 meters and a height of 3 meters. The top of the tank is a hemisphere with a radius of 2 meters. What is the total volume of the water tank? (3 Marks)
Ans: To find the total volume of the water tank, we need to find the volumes of the cylindrical base and the hemispherical top and then add them together.
The volume of the cylindrical base is given by:Volume of cylinder = πr2h Where r is the radius of the base and h is the height of the cylinder.
Substituting the given values, we get:Vcyl = π.(2)2.(3) = 12π cubic meters
The volume of the hemispherical top is given by:Vhemi = 2/3 πr3 where r is the radius of the hemisphere.
Substituting the given value, we get:Vhemi = 2/3 π.(2)3 = 16/3π cubic meters
Therefore, the total volume of the water tank is:Vtotal = Vcyl + Vhemi = 12π + 16/3π = 44/3π cubic meters ≈ 46.15 cubic meters
Hence, the total volume of the water tank is approximately 46.15 cubic meters.
Ques: A rectangular prism with dimensions 10 cm x 5 cm x 4 cm is glued to the top of a rectangular prism with dimensions 6 cm x 5 cm x 4 cm. What is the total volume of the combined solid? (2 Marks)
Ans: The total volume of the combined solid can be found by adding the volumes of the two rectangular prisms.
Volume of the first rectangular prism = length x width x height = 10 cm x 5 cm x 4 cm = 200 cm3
Volume of the second rectangular prism = length x width x height = 6 cm x 5 cm x 4 cm = 120 cm3
To find the total volume, we add these two volumes:
Total volume = 200 cm3 + 120 cm3 = 320 cm3
Therefore, the total volume of the combined solid is 320 cm3.
Ques: A cone with a radius of 4 cm and a height of 6 cm is placed on top of a cylinder with a radius of 3 cm and a height of 10 cm. What is the total volume of the combined solid? (3 Marks)
Ans: The volume of the cone is given by:Vcone = (1/3) * π * r2 * h where r is the radius and h is the height of the cone.
Substituting the given values, we get:Vcone = (1/3) * π * 42 * 6 Vcone = 32π/3
The volume of the cylinder is given by:Vcylinder = π * r2 * h wherer is the radius and h is the height of the cylinder.
Substituting the given values, we get:
Vcylinder = π * 32 * 10 Vcylinder = 90π
The total volume of the combined solid is the sum of the volumes of the cone and the cylinder:Vtotal = Vcone + Vcylinder Vtotal = 32π/3 + 90π Vtotal = 186π/3 Vtotal = 62π
Therefore, the total volume of the combined solid is 62π cubic centimeters.
Ques: A wooden block in the shape of a rectangular prism has dimensions 10 cm x 5 cm x 3 cm. A cylindrical hole with a radius of 1 cm is drilled through the center of the block, such that the axis of the hole is perpendicular to the larger faces of the block. What is the volume of the remaining wooden block? (3 Marks)
Ans: The volume of the wooden block is: 10 cm x 5 cm x 3 cm = 150 cubic cm
The cylindrical hole passes through the center of the block, so it divides the block into two identical halves. The volume of the cylindrical hole is: π x 12 x 3 cm = 3π cubic cm
Thus, the volume of the remaining wooden block is:150 cubic cm - 2(3π cubic cm) = 150 cubic cm - 6π cubic cm= 150 - 18.85= 131.15 cubic cm (rounded to two decimal places)
Therefore, the volume of the remaining wooden block is approximately 131.15 cubic cm.
Ques: A sphere with a radius of 6 cm is placed inside a cylindrical container with a radius of 8 cm and a height of 12 cm, such that the sphere rests on the bottom of the container. What is the volume of water needed to fill the container and the sphere? (3 Marks)
Ans: To find the volume of water needed to fill the container and sphere, we need to find the combined volume of the sphere and the cylinder.
The volume of the sphere is given by the formula: Vsphere = (4/3)πr3 where r is the radius of the sphere.
Substituting the given values, we get:Vsphere = (4/3)π(6)3 Vsphere = 904.78 cm3 (rounded to two decimal places)
The volume of the cylinder is given by the formula: Vcylinder = πr2 hwhere r is the radius of the cylinder and h is the height of the cylinder.
Substituting the given values, we get:Vcylinder = π.(8)2(12) Vcylinder = 2410.68 cm3 (rounded to two decimal places)
Therefore, the combined volume of the sphere and the cylinder is:Vcombined = Vsphere + Vcylinder Vcombined = 904.78 + 2410.68 V combined = 3315.46 cm3 (rounded to two decimal places)
So, the volume of water needed to fill the container and the sphere is 3315.46 cubic centimeters.
Ques: What is the Volume of a Solid? (2 Marks)
Ans: The volume of a solid is the amount of space occupied by an item. It is determined by the number of unit cubes required to fill the solid.
Ques: Find the volume of the cube whose each edge is 5cm. (2 Marks)
Ans: From the given edge a = 5cm
The volume of a cube is a3.
V = a3 cubic units
⇒V = 53
⇒V = 125cm3
Hence, the Volume of a Cube is 125cm3.
Ques: The volume of a cylindrical water tank is 1000m3, and the radius of the base of the cylindrical tank is 5m. Calculate the height of the tank. (2 Marks)
Ans: Given: Radius (r) of base =5m
and the volume of the tank =1000m3
We know that the volume of a cylinder = πr2h
⇒1000=22/7×5×5×h
⇒h=12.7m
Hence, the height of the tank is 12.7m.
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