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A Pyramid is a polyhedron, which is a three dimensional solid usually joined at their edges. A pyramid can be of many types like a rectangular pyramid, pentagonal pyramid, a square pyramid, and hexagonal pyramids. Just like these types, a triangular pyramid has a polygon base (of any number of sides) and lateral faces of which are triangles with a common vertex. A pyramid is named after its base thus, a triangular pyramid has a triangle as its base. A triangular pyramid formula consists of both the volume and the surface area of the pyramid that calculates - the three triangular shaped sides, the height, and the slant height.
Also Read: Solids Shapes
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Key Terms: Triangular pyramid, Triangular pyramid formula, Apex, height, surface area, base area, tetrahedron
What is a Triangular Pyramid?
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A pyramid is formed when a base is constructed to connect to the apex. A triangular based pyramid or triangular pyramid is also known as Tetrahedron. The triangular pyramid formula includes both the volume and the surface area of the pyramid. The formula calculates the base area and the height whereas the surface area of the triangular pyramid calculates the base area, perimeter, and slant height. The slant height is the diagonal height from the center of one of the base edges to the apex. Height ‘h’ is the perpendicular distance from the apex to the base.
Apex of the pyramid
Read: Volume of a square Pyramid
Triangular Pyramid Formula
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There are two formulas for triangular pyramid:
Volume of a triangular pyramid =1/3 x Base Area X Height
Surface Area of triangular pyramid = Base Area + ½ (perimeter x slant height)
Triangular Pyramid
Read: Volume of Pyramid
Surface Area
When side faces are the same:
Surface Area = [Base Area] + ½ x Perimeter x [Slant Length]
For example: Base Area is 25, Perimeter is 18, Slant Length is 4
Using the above formula,
= 25 + ½ x 18 x 4
= 25 + 36
= 61
Note: If side faces are different then calculate the area of the base and each triangular face separately and then add them up.
Volume
Volume = 1/3 x [Base Area] x Height
For example: Base Area is 28, height is 4.5
Using the above formula,
= 1/3 x 28 x 4.5
= 1/3 x 126
= 42
Things to Remember
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- All sides and faces of a triangular pyramid are the same. When rotated or placed, it will look the same.
- The top of the pyramid is called apex. It is where the edges meet in the middle.
- A triangular pyramid has 6 edges, 4 faces, and 4 vertices.
- A triangular pyramid is also known as the tetrahedron.
- A tetrahedron having equilateral triangles for each of its faces is known as a regular tetrahedron.
- A tetrahedron having edges of different length is known as an irregular tetrahedron.
Sample Questions
Ques. Find the volume of a triangular pyramid having base area of 10 cm2 and height of 5 cm. (3 marks)
Ans. Base area = 10cm2, height = 5cm …given,
By using the formula of volume of triangular pyramid
Volume= 1/3 x Base area x height
= 1/3 x 10 x 5
= 16.6 cm3
Ques. A triangular pyramid has a base area of 15 units2 and a sum of the lengths of the edges 60 units. Calculate the surface area of the triangular pyramid if the slant height is 20 units. (3 marks)
Ans. Base area = 15 units2, perimeter= 60 units …given,
By using the formula for surface area of triangular pyramid,
Surface area = Base area + ½ (perimeter x slant height)
= 15+½ (60x20)
=615 unit2
Ques. Find the surface area of a triangular pyramid whose area of the base triangle is 24 unit2, the perimeter of the triangle is 12 units, and the slant height of the pyramid is 18. (3 marks)
Ans. The area of the base triangle = 24 unit2
Perimeter of the triangle = 12 units
Slant height of the pyramid = 18 units
By using the formula for the surface area of triangular pyramid,
The surface area of a triangular pyramid = Base Area + ½ (Perimeter x slant height)
= 24 + ½ (12x18)
= 132 unit2
Ques. What is a net of triangular pyramids? (2 marks)
Ans. The net of a triangular pyramid is made up of 4 triangles when laid flat including the base and three lateral faces.
Net of a triangular pyramid
Ques. What are the properties of a triangular pyramid? (3 marks)
Ans. 1. A triangular pyramid has 4 faces, 4 vertices, 6 edges.
- Unlike any other pyramid, the base is also triangular in shape.
- Triangular pyramids can be regular, irregular or non-regular, and right angled.
Ques. What are the types of triangular pyramids? (3 marks)
Ans. There are three types of triangular pyramids:
- Regular - triangular pyramid having a base with sides equal in length. In the case of a triangular pyramid, the base of a regular tetrahedron should be an equilateral triangle.
- Irregular - Irregular or non-regular pyramid is the one having a base with all sides of different lengths. Even the base of an irregular pyramid has unequal sides. The base of a non-regular triangular pyramid could either be a scalene or an isosceles triangle.
- Right-angled - A right angled triangular pyramid having a right-angle triangle as its base jutting up to a single point.
Ques. What if two congruent triangular pyramids are stuck together along their bases. How many faces, edges, and vertices does this bipyramid have? (2 marks)
Ans. Sticking together two triangular pyramids along their bases form a triangular bipyramid.
Sectional view of bipyramid
Above bipyramid has 6 triangular faces, 9 edges and 5 vertices.
Ques. How do you recognize a solid as a polyhedron? (2 marks)
Ans. A solid is said to be a polyhedron if it is made up of only polygonal-faces, the faces meet at the edge with one line segment and the edges meeting at a point. The meeting point is called a vertex.
Ques. What do you mean by lateral surface area of a triangular pyramid? (2 marks)
Ans. Lateral surface area is the area of non-base faces and it is calculated by removing the base area of an object. The lateral surface is calculated by removing the base area of triangle from the product of the perimeter of the base and slant height of a pyramid.
Ques. Verify Euler relation with triangular pyramid. (3 marks)
Ans. A triangular has,
Face = 4
Vertices = 4
Edges = 6
=> F+V = E+2
=> F+V-E = 2
This relationship is called Euler’s Formula
=> 4+4-6 =2
=> 8-6 =2
=> 2=2
Therefore, LHS = RHS
Hence, Verified.
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