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Pyramid is a polyhedron with a polygonal base and triangles for sides. The pyramid formula is:
- Surface Area of Pyramid = Surface Area of a Pyramid = Base Area + 1/2 (Number of Base Sides) x Slant Height x Length of Base
- Volume of Pyramid = \(\frac{1}{3}\) x Base area x Height
A pyramid is formed by the combination of three parts, firstly the ‘base’ from which the ‘faces’ of the pyramid originates. These faces are triangular sides of the pyramid and they meet together at a point on top of the pyramid known as the ‘apex’ of the pyramid. A pyramid is named after its base.
Also Check: 3-D Shapes
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Key Terms: Pyramid, Surface Area, Volume, Base Length, Apex, Area, Polyhedron
What is a Pyramid?
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A pyramid is a polyhedron with a polygonal base and triangles for sides. A pyramid can be formed by combining three parts, which is the ‘base’ from which the ‘faces’ of the pyramid originate. The three parts of a pyramid are:
- Apex
- Face
- Base
General formula for determining:
- Surface area of Pyramid: base area + \(\frac{1}{2}\)(no. of base sides + slant height × Base Length
- Volume of Pyramid: \(\frac{1}{3}\) × base area × height
Types of Pyramids
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The types of the pyramid are as listed below:
- Square pyramid
- Triangular pyramid
- Pentagonal pyramid
- Hexagonal Pyramid
- Right Pyramid
- Oblique Pyramid
Square Pyramid
In a square pyramid, the base of the pyramid is a square, and the four triangular sides originate from the 4 edges of the square base, and all these triangular sides meet at a point called the apex.
Triangular Pyramid
In a triangular pyramid, the base of the pyramid is a triangle with 3 triangular faces.
Pentagonal Pyramid
In a pentagonal pyramid, the base is a pentagon, and it has 5 triangular faces.
Hexagonal Pyramid
In a hexagonal pyramid, the base of the pyramid is a base, and there are 6 adjoining triangular faces.
Right Pyramid
A right pyramid can be defined as a pyramid in which the apex lies directly above the base’s centroid.
Oblique Pyramid
An Oblique Pyramid can be defined as a type of pyramid where the apex is not over the midpoint of its base.
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Pyramid Formula
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The various types and formula of pyramids are as shown below:
Pyramid Formulas
In all these above-mentioned formulas of pyramid, the terms ‘a’,’b’,’s’,’h’, represents the following terms:
- a = apothem length of pyramid
- b = base length of the pyramid
- s = slant height of the pyramid
- h = height of the pyramid
Apothem length of Pyramid
It is a line segment from the centre to the midpoint of one of its side.
Base length of pyramid
The Base Length of a pyramid is the length of any one edge of the base.
Slant Height of Pyramid
It is the distance measured along a lateral side from the base to the apex along the centre face.
Height of the pyramid
It is the perpendicular distance between the apex and the centre of the base.
Read More: Surface Area of Cuboid
Things to Remember
- A pyramid is named after its base. If the base of the pyramid is triangle, it is called a triangular pyramid and so on.
- A pyramid is a combination of a polygon base and triangular sides.
- In any pyramid, if the number of vertices are ‘n’, then: Number of edges= 2n - 2, and Number of triangular faces= n - 1
Read More: Area of Pentagon
Sample Questions
Ques. What is the surface area of a pentagonal pyramid if the following values are given;
a = 6cm
b = 9cm
s = 2cm (2 marks)
Ans. Surface area of pentagonal pyramid is = \( \frac{5}{2}\)ab + \( \frac{5}{2}\)bs
= \( \frac{5}{2}\)6.9 + \( \frac{5}{2}\)9.2
= 15 × 9 + 45
= 190 cm2
Ques. What is a pyramid and what are its main components? (2 marks)
Ans. a pyramid is a polyhedron, whose base is a polygon , and whose lateral faces are triangles with a common vertex.
The three main component of a pyramid are:
- Base
- Face
- Apex
Ques. what is the surface area of square pyramid with, base length 4cm, slant height 10 cm and height 8cm? (2 marks)
Ans. surface area of square pyramid= 2bs+b2
Given values are; b = 4cm
S = 10cm
Now putting these values in surface area formula;
2bs + b2 =2.4.10 + 16
= 80 + 16= 96
So the surface area of the pyramid is 96 cm2.
Ques. What is an hexagonal pyramid? How many edges,vertices and faces are there in a hexagonal pyramid? (2 marks)
Ans. A hexagonal pyramid is that kind of pyramid, in which the base of the pyramid is a hexagon, and the sides are triangles. In an hexagonal pyramid, there are 7 vertices, 6 triangular faces and 12 edges.
Ques. How many faces, edges and vertices do a pyramid have? (3 marks)
Ans. In a pyramid, the base is a polygon, and the faces are triangles, we can calculate the number of faces, edges and vertices by checking the diagram for that pyramid. Let us take an example of a triangular pyramid.
In this diagram, ABC is the base. ACD, ABD, CDB are the three triangular faces. A,B,C,D are the 4 vertices. AB,AC,BC,AD,BD,CD are the 6 vertices.
In general, if the number of vertices is ‘n’, the number of edges will be ’2n-2’and number of triangular faces will be ‘n-1’
Ques. If the volume of the square pyramid is 121cm3, and the height of the pyramid is 9cm, calculate the base length of this given pyramid. (3 marks)
Ans. Volume of square pyramid = \(\frac{1}{3}\)b2h
Given values are;
Volume = 121cm3
Height = 9cm
Base length=?
Now, putting these values in the volume formula we will get;
\(\frac{1}{3}\)b2h=121
9b2=363
b2 = 40.33
b = 6.350cm
Ques. What is the volume of a pentagonal pyramid, if the apothem length of the pyramid is 12cm, base length is 12cm and height of the pyramid is 15 cm. (3 marks)
Ans. Volume of pentagonal pyramid= \(\frac{5}{2}\)abh
The given values are; a= 12cm
b = 12cm
h = 15cm
putting all these values in pentagonal pyramid formula;
volume= \(\frac{5}{2}\)×12×12×15
= 5400cm3
Ques. What is the surface area of a hexagonal pyramid, if its base area is 29cm2, apothem length is 2 cm and slant height is 10 cm? (3 marks)
Ans. As per the question,
base area= 3ab
29 = 3.2.b
b = 4.83
now the surface area = 3ab + 3bs
3 × 2 × 4.83 + 3 × 4.83 × 10
= 28.98+144.9
=173.88 cm2
Ques. Find the apothem length of the triangular pyramid, If its base area is 27cm2 and base length is 3 cm? (3 marks)
Ans. The base area of triangular pyramid= \(\frac{1}{2}\)ab
Now putting the given values in this formula, we get
27 = \(\frac{1}{2}\)a3
a = \(\frac{27.2}{3}\)
a = 18 cm
Ques. What is the volume of the square pyramid, with the height of 3.5 cm, and base area of 6cm? (3 marks)
Ans. Volume of square pyramid is =\(\frac{1}{3}\)b2h
The given values are h=3,5cm
b =6cm
now putting all these values in the volume formula we get
\(\frac{1}{3}\)b2h=\(\frac{1}{3}\)(36)(3.5)
Volume= 21 cm3
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