In Geometry, a pyramid is a three-dimensional shape whose base should be a polygon, and its face should be a triangle. All the triangular side faces meet at a common point called the vertex or apex. The altitude or height of the pyramid is the perpendicular distance from the apex to the centre of the base. Whereas, slant height is a measure of the perpendicular drawn to the base of the side face of a triangle from the apex. In this article, we are going to discuss the surface area of pyramid formulas and examples in detail.
What is the Surface Area of a Pyramid?
The surface area of a pyramid is defined as the sum of the areas of all faces of a pyramid. In other words, the surface area of a pyramid is the total area occupied by the surface of the pyramid. It is generally measured in square units, such as m2, cm2, in2, and so on.
The two types of the surface area of the pyramid are:
- Lateral Surface Area (LSA of Pyramid) = Sum of the area of the side faces of a pyramid.
- Total Surface Area (TSA of Pyramid) = LSA of pyramid + Area of the base.
General Formula for Surface Area of Pyramid
The general formula for the surface area of the pyramid is as follows:
The lateral surface area of the regular pyramid formula is given by:
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Lateral Surface Area of Regular Pyramid = (½) Pl Square units |
Similarly, the total surface area of the regular pyramid formula is given by:
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Total Surface Area of Regular Pyramid = (½)Pl + B Square units |
Where “l” is the slant height of a pyramid
“P” is the base perimeter of a pyramid
“B” represents the base area of a pyramid
Surface Area of Pyramid Formulas
Generally, if we are asked to find the surface area of the pyramid without any specifications, it represents the total surface area. Now, let us discuss the surface area of pyramid formulas with different bases.
Triangular Pyramid
In the triangular pyramid, the base of the pyramid is a triangle.
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Surface Area of Triangular Pyramid = 1⁄2(a × b) + 3⁄2(b × s) square units |
Where, b = side length, a = height, s = slant height.
Square Pyramid
In the Square pyramid, the base of the pyramid is a square.
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Surface Area of Square Pyramid = 2bs + b2 Square units |
Where, b = side length, s = slant height.
Pentagonal Pyramid
In the pentagonal pyramid, the base of the pyramid is a pentagon.
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Surface Area of Pentagonal Pyramid = 5⁄2(a × b) + 5⁄2(b × s) Square units |
Where, b = side length, s = slant height, a = apothem length.
Hexagonal Pyramid
In the hexagonal pyramid, the base of the pyramid is a hexagon.
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Surface Area of Hexagonal Pyramid =3(a × b) + 3(b × s) Square units |
Where, b = side length , s = slant height, a= apothem length.
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Surface Area of Pyramid Examples
Go through the solved examples on the surface area of the pyramid:
Example 1:
Determine the surface area of the square pyramid, given that side length = 4 cm and slant height = 8 cm.
Solution:
Given: Side length, b = 4 cm
Slant height, s = 8 cm
We know that the surface area of square pyramid = 2bs + b2 Square units
Now, substitute the known values in the formula
S.A = 2(4)(8) + (4)2
S.A = 64+16 = 80
Hence, the surface area of the square pyramid is 80 cm2.
Example 2:
Compute the base area of the pentagonal pyramid, given that side length = 6.4 m and apothem = 16 m.
Solution:
Given: Side length, b = 6.4 m
Apothem = 16 m
We know that the base of a pentagonal pyramid is a pentagon. Hence, base area = 5⁄2(a × b) square units.
Base area = (5/2)(16×6.4)
Base area = 5(8×6.4) = 5(51.2) = 256
Thus, the base area of a pentagonal pyramid is 256 m2.
Practice Questions
Solve the following problems:
- Determine the surface area of the triangular pyramid, given that side length = 2 cm, height = 4 cm and slant height = 5 cm.
- Compute the base area of the square pyramid whose side length is equal to 7 cm.
- Find the surface area of the square pyramid given that side length = 5 cm and slant height = 7 cm.
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Frequently Asked Questions on Surface Area of Pyramid
What is the surface area of a pyramid?
The surface area of the pyramid is the total area occupied by the surface of the pyramid.
What is the general formula for the lateral surface area of a regular pyramid?
If “P” is the base perimeter and “l” is the slant height, then the lateral surface area of the regular pyramid formula is given by (½) Pl Square units.
What is the general formula for the total surface area of a regular pyramid?
If “P” is the base perimeter, “l” is the slant height, and “B” is the base area, then the total surface area of the regular pyramid formula is given by (½)Pl + B Square units.
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