Surface Area Of Pyramid - Formula, Definition, And Examples

Surface Area of Pyramid

The surface area of a pyramid is obtained by adding the area of all its faces. A pyramid is a three-dimensional shape whose base is a polygon and whose side faces (that are triangles) 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.

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 all its faces. Observe the pyramid given below to see all its faces and the other parts like the apex, the altitude, the slant height, and the base.

The surface area of a pyramid is 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 the 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 refers to the total surface area of the pyramid.

Surface Area of Pyramid Formula

The surface area of a pyramid can be calculated by finding the areas of each of its faces and adding them. 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 'l'. Then,

• The Lateral Surface Area of pyramid (LSA) = (1/2) Pl
• The Total Surface Area of pyramid (TSA) = LSA + base area = (1/2) Pl + B

Note that we will use the formulas for the area of polygons to calculate the base areas here. Now, let us see how to derive the formulas of the surface area of a pyramid.

Proof of Surface Area of Pyramid Formula

The surface area of a pyramid involves 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 'a' and whose slant height is 'l'.

Then,

• The base area (area of square) of the pyramid is, B = a2
• The base perimeter (perimeter of square) of the pyramid is, P = 4a
• The area of each of the side faces (area of triangle) = (1/2) × base × height = (1/2) × (a) × l

Therefore, the sum of all side faces (sum of all 4 triangular faces) = 4 [(1/2) × (a) × l] = (1/2) × (4a) × l = (1/2) Pl. (Here, we replaced 4a with P which represents its perimeter.)

Hence, the Lateral Surface Area of the pyramid (LSA) = (1/2) Pl

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) Pl + B

Using these two formulas, we can derive the surface area formulas of different types of pyramids.

Surface Area of Pyramid with Altitude

The surface area of a pyramid can be calculated if its altitude is given. Observe the figure given below which shows that the triangle formed by half the side length of the base (a/2), the slant height (l), and the altitude (h) is a right-angled triangle. Hence, we can apply the Pythagoras theorem and find out the slant height if the altitude and base length is given. Thus, l2 = h2 + (a/2)2

So, we can calculate the slant height using the formula, l2 = h2 + (a/2)2. Now that we have the slant height, the base length, and the height, we can find the surface area of the pyramid using the formula, Total surface area of pyramid (TSA) = LSA + base area = (1/2) Pl + B

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