how to find surface area of a pyramid

Surface Area of Pyramid

The surface area of a pyramid is obtained by adding the areas of its faces. A pyramid is a three-dimensional shape whose base is a polygon and whose side faces (that are triangles) all meet at a point which is called the apex (or) vertex. The perpendicular distance from the apex to the center of the base is called the altitude or height of the pyramid. The length of the perpendicular drawn from the apex to the base of a triangle (side face) is called the "slant height". Let us learn more about the surface area of a pyramid along with its formula, a few solved examples, and practice questions.

Parts of a Pyramid

1.	What is the Surface Area of Pyramid?
2.	Surface Area of Pyramid Formula
3.	Proof of Surface Area of Pyramid Formula
4.	FAQs on Surface Area of Pyramid

What is the Surface Area of Pyramid?

The surface area of a pyramid is a measure of the total area that is occupied by the surface. In other words, it's the sum of areas of its faces and hence it is measured in square units such as m², cm², in², ft², etc. A pyramid has two types of surface areas, one is lateral surface area (LSA) and the other is the total surface area (TSA).

The lateral surface area (LSA) of a pyramid = The sum of areas of the side faces (triangles) of the pyramid.
The total surface area (TSA) of a pyramid = LSA of pyramid + Base area.

In general, the surface area of a pyramid without any specifications means the total surface area of the pyramid.

TSA and LSA of Pyramid

Surface Area of Pyramid Formula

The surface area of any pyramid can be found by finding the areas of each of its faces and adding them. But if the pyramid is regular (i.e., a pyramid whose base is a regular polygon and whose altitude passes through the center of the base), there are some specific formulas to find the lateral surface area and total surface area. Consider a regular pyramid whose base perimeter is 'P', the base area is 'B', and slant height (the height of each triangle) is 's'. Then,

The lateral surface area of pyramid (LSA) = (1/2) Ps
The total surface area of pyramid (TSA) = LSA + base area = (1/2) Ps + B

surface area of pyramid formula

Note that we will have to use the area of polygons formulas to calculate the base areas here. How to derive the formulas of the surface area of a pyramid? Let us see.

Proof of Surface Area of Pyramid Formula

Why does the surface area of the pyramid involve the perimeter and slant height? Let us understand the formulas of LSA and TSA of a pyramid by taking a specific pyramid as an example. Let us consider a square pyramid whose base length is 'x' and whose slant height is 's'.

Proof of Surface Area of Pyramid Formula

Then,

The base area (area of square) of the pyramid is, B = x²
The base perimeter (perimeter of square) of the pyramid is, P = 4x
The area of each of the side faces (area of triangle) = (1/2) × base × height = (1/2) × (x) × s

Thus, the sum of all side faces (sum of all 4 triangular faces) = 4 [(1/2) × (x) × s] = (1/2) × (4x) × s = (1/2) Ps.

Hence, the lateral surface area of the pyramid (LSA) = (1/2) Ps.

We know that the total surface area of a pyramid (TSA) is obtained by adding the base and lateral surface areas. Thus,

The total surface area of pyramid (TSA) = LSA + base area = (1/2) Ps + B

Using these two formulas, we can derive the surface area formulas of different types of pyramids. In specific, you can find the formulas for the surface area of a triangular pyramid and the surface area of a square pyramid here.

Note: The triangle formed by half the side length of the base (x/2), the slant height (s), and the altitude (h) is a right-angled triangle and hence we can apply the Pythagoras theorem for this. Thus, (x/2)² + h² = s². We can use this while solving the problem of finding the surface area of the pyramid given its altitude.

Altitude, Base Length, and Slant Height of a Pyramid

Examples on Surface Area of Pyramid

Example 1: Calculate the lateral surface area of a square pyramid if the side length of the base is 14 inches and the slant height of the pyramid is 20 inches.

Solution:

The side length of the base, x = 14 inches

Then, the perimeter of the base (square) is, P = 4x = 4(14) = 56 inches.

Slant height, s = 20 inches

The lateral surface area of a square pyramid is

Lateral surface area (LSA) = (1/2) Ps

= (1/2) × (56) × 20

= 560 in²

Therefore, the lateral surface area of the given pyramid is 560 in².
Example 2: Find the base area of a pentagonal pyramid whose base length (side length of the regular pentagon) is 6.4 units and whose apothem is 16 square units.

Solution:

Given, the side length of the pentagon is, s = 6.4 units and its apothem is, a = 16 units.

The base area (B) of the pentagonal pyramid is nothing but the area of the regular pentagon. Thus, B = (5/2) sa

= (5/2) × 6.4 × 16

= 256 square units.

Therefore, the base area of the given pentagonal pyramid = 256 square units.
Example 3: Calculate the total surface area of a square pyramid if the side of the base is given as 14 inches and the height (altitude) of the pyramid is given as 24 inches.

Solution:

Given, the side of the base is, x = 14 inches, and the height of the pyramid is, h = 24 inches.

Let its slant height be 's'.

By Pythagoras theorem, s² = (x/2)² + h²

s² = (14/2)² + 24² = 625

Thus, s = 25 inches.

The perimeter of the base is, P = 4x = 4(14) = 56 inches.

The base area is, B = 14² = 196 square inches.

The total surface area of the square pyramid is,

TSA = (1/2) Ps + B

TSA = (1/2) × 56 × 25 + 196 = 896 square inches.

Therefore, the TSA of the given pyramid = 896 square inches.

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Practice Questions on Surface Area of Pyramid

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FAQs on Surface Area of Pyramid

What is the Definition of Surface Area of Pyramid?

The surface area of a pyramid is defined as the sum of areas of its faces. There are two types of surface areas: the total surface area (TSA), the sum of the areas of all the faces, and the other is the lateral surface area (LSA), which is the sum of the areas of the sides faces.

What is the Total Surface Area of Pyramid?

The total surface area of a pyramid is obtained by adding the areas of all its faces (both the base and the side faces). The total surface area of a pyramid whose base perimeter is 'P', the base area is 'B', and slant height is 's' is TSA = (1/2) Ps + B.

What is the Lateral Surface Area of Pyramid?

The lateral surface area of a pyramid is the sum of the areas of all its side faces (which are triangles). The lateral surface area of a pyramid whose perimeter of the base is 'P' and slant height 's' is LSA = (1/2) Ps.

What is the Formula for Surface Area of Pyramid?

There are two types of surface areas of a pyramid, one is the total surface area and the other is the lateral surface area. Let us consider a pyramid of base area 'B', slant height 's', and base perimeter 'P'. Then:

Total surface area = (1/2) Ps + B.
Lateral surface area = (1/2) Ps.

How to Find Surface Area of Pyramid With Slant Height?

Consider a pyramid whose slant height is 's', the base perimeter is 'P', and the base area is 'B'. Its total surface area can be calculated using the formula, (1/2) Ps + B. The base area can be found by applying the formulas of the area of a polygon. (can we write in steps?)

How to Find Surface Area of Pyramid With Height (or Altitude)?

Let us consider a pyramid whose base is a regular polygon of side length 'x'. Assume that the slant height of the pyramid is 's' and its altitude is 'h'. Also, assume that only 'x' and 'h' are given and you are asked to find the surface area. Then:

Step 1: Find 's' using the relation (by Pythagoras theorem), s² = (x/2)² + h².
Step 2: Find the base perimeter 'P'.
Step 3: Find the base area 'B'.
Step 4: Find the total surface area of the pyramid using the formula (1/2) Ps + B.