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A pyramid is a three-dimensional shape whose base is made up of a polygon that is connected to a point known as its apex. Here we will learn how to calculate the surface area of pyramids. Hint: Think about the shapes that make up the sides of a pyramid....Read MoreRead Less

The history of the pyramids is very old. The ancient Egyptians first built pyramids around 2700 BC until 1700 BC. The pyramids are huge structures that are built mainly with bricks and stones. The word pyramid came from the Greek word ‘pyramids’. The pyramids are the largest constructions in the world. The world’s largest pyramid by volume is the great pyramid of Cholula.

The pyramids of Egypt are in the list of Seven Wonders of the World.

The outer surface of the pyramid is triangular and the surface gradually makes a single sharp top. The base of pyramids are triangle, quadrilateral or any polygon. The pyramids of Egypt are in the list of Seven Wonders of the World.

A pyramid is a three dimensional shape or structure whose base is a polygon and all the lateral surfaces are triangular. The triangular faces meet at the top and make a vertex. This vertex is called the apex of a pyramid. If there are N number of sides in the base of the pyramid then the numbers of faces will be N + 1, the number of edges will be 2N and the number of vertices will be N + 1 in that pyramid.

The height of the pyramid is the perpendicular distance from the apex to the base of the pyramid. The height of the triangular faces is called the slant height of the pyramid. If the base of the pyramid is a regular polygon then the pyramid is said to be a regular pyramid. Some different types of pyramid based on the shape of its base are triangular pyramid, square pyramid, hexagonal pyramid and so on.

The surface area (S) of a pyramid is the sum of the areas of the base and the lateral faces. The area of the base and lateral triangles can be found separately using respective formulas for the area of a polygon. Then, the results are added to get the total surface area of a pyramid.

S = area of base + area of lateral faces

If the base of a pyramid is in the shape of a rectangle then the pyramid is called the rectangular pyramid. To find the area of a rectangular pyramid the rectangular base area and the areas of the four triangular faces are required. The area of the base and the lateral surfaces are later added to calculate the area of the rectangular pyramid.

The slant height of width face of the pyramid is \(~=~\sqrt{\left(\frac{l}{2}\right)^2~+~h^2}\)

The slant height of length face of the pyramid is \(~=~\sqrt{\left(\frac{w}{2}\right)^2~+~h^2}\)

Therefore the area of the rectangular pyramid formula is,

S = area of base + area of lateral faces

\(S~=~l~\times~\ w~+~2~\times~\left(\frac{1}{2}~\times~ w~\times~\sqrt{\left(\frac{l}{2}\right)^2~+~h^2}\ \right)~+~2~\times~\left(\frac{1}{2}~\times~ l~\times~\sqrt{\left(\frac{w}{2}\right)^2~+~h^2}\ \right)\)

\(S~=~l~\times~\ w~+~w~\times~\sqrt{\left(\frac{l}{2}\right)^2~+~h^2}~+~l~\times~\sqrt{\left(\frac{w}{2}\right)^2~+~h^2}\)

This formula is a little complicated for use. The manual calculation is more preferable for finding the area of a rectangular pyramid.

If the base of a pyramid is in the shape of a triangle then the pyramid is called the triangular pyramid. To find the area of a triangular pyramid the triangular base area and the areas of the three triangular faces are required. The area of the base and the lateral surfaces are later added to calculate the area of the square pyramid.

S = area of base + area of lateral faces

S = \(1/2 ~\times~ \text{base} ~\times~ \text{height} ~+~1/2 ~\times ~\text{perimeter} ~\times ~\text{slant height}\)

S = \(1/2\ ~\times~ b ~\times~\ h +~1/2\ ~\times~\ P~\times~\ L\)

This formula is also a little complicated for use. The manual calculation is more preferable for finding the area of a square pyramid.

If the base of a pyramid is in the shape of a square then the pyramid is called a square pyramid. To find the area of a square pyramid the square base area and the areas of the four triangular faces are required. The area of the base and the lateral surfaces are later added to calculate the area of the square pyramid.

S = area of base + area of lateral faces

The slant height of the pyramid is \(~=~\sqrt{\left(\frac{a}{2}\right)^2~+~h^2}\)

S = area of base + area of lateral faces

S = \(a^2~+~4\left(\frac{1}{2}~\times~ a~\times~\sqrt{\left(\frac{a}{2}\right)^2~+~h^2}\right)\)

S = \(a^2~+~2a~\times~\sqrt{\left(\frac{a}{2}\right)^2~+~h^2}\)

This formula is also a little complicated for use. The manual calculation is more preferable for finding the area of a square pyramid.

**Example 1: **Find the surface area of a regular pyramid shown in the figure. The base of the pyramid is a square of side 5 inches and the slant height is 8 inches.

**Solution:**

First draw a net figure for the areas of the pyramid.

The area of the base can be obtained using the area formula of square. the area of four lateral faces can be found by multiplying the area of each triangle by 4.

Area of base = 5 x 5 [Area of square formula]

= 25 [Simplify]

Area of a lateral face \( ~=~\frac{1}{2}~\times~5~\times~8\)

\( ~=~ 20\)

Now, find the sum of the area of the base and the lateral faces.

S = area of base + area of lateral faces

= 25 + 4 x 20 [There are four congruent lateral faces]

= 105 [Simplify]

The surface area of the pyramid is about 105 square inches.

**Example 2: **Find the surface area of a regular pyramid shaped roof top of a house as shown in the figure. The base of the pyramid is a square of side 5 inches and the slant height is 8 inches**.**

**Solution:**

First draw a net figure for the areas of the pyramid.

The area of the base can be neglected because the base is inside the house and the base surface is not visible.

Area of a lateral face \(~=~\frac{1}{2}~\times~10~\times~12\) [Area of triangle formula]

= 60

Therefore the area of four lateral triangles = 4 × 60 = 240

The surface area of the pyramid roof top is about 240 square feet.

**Example 3: **Find the surface area of the regular pyramid shown in figure.

**Solution:**

First draw a net figure for the areas of the pyramid.

The area of the base can be obtained using the area formula for a triangle. Area of three lateral faces can be found by multiplying the area of each triangle by 3.

Area of base = \(\frac{1}{2} ~\times~ 10~ \times ~8.7\) [Area of a triangle formula]

= 43.5 [Simplify]

Area of a lateral face = \(\frac{1}{2} ~\times~ 10 ~\times ~14\) [Area of triangle formula]

= 70

Now, find the sum of the area of the base and the lateral faces.

S = area of base + area of lateral faces

= 43.5 + 3 x 70 [There are three congruent lateral faces]

= 253.5 [Simplify]

The surface area of the pyramid is about 253 . 5 square inches.

**Example 4: **Find the surface area of a triangular pyramidal gift box. Every face of the box is in the shape of an equilateral triangle as shown in the figure.

**Solution:**

All the four faces of the gift box is an equilateral triangle. Therefore the area of the pyramid can be calculated using the area of a face multiplied by 4.

Area of each triangular face \( ~= \sqrt3/4\ a^{2\ }\ \) [Area of an equilateral triangle formula]

= \( \sqrt3/4\ {(6)}^{2\ }\) [Replace a with 6]

\(\approx~15.60\) [Simplify]

Therefore the surface area of gift box

= 62.4

The surface area of the gift box is about 62.4 square inches.

**Example 5: **The roof of a house is shaded like a square pyramid. If one bundle of shingles covers 27 square feet, How many bundles should you buy to cover the rooftop? The measurements of the pyramidal roof is shown in figure.

**Solution:**

** **

The roof is like a square pyramid. The base of the pyramid does not need the shingles. So, only find the sum of the lateral faces of the roof. Then divide the area of roof by the area of each shingle to find the number of shingles required.

Area of lateral face \(~ =~ \frac{1}{2}~\times~18~\times~15\) [Area of triangle formula]

= 135

There are four identical faces. So, the lateral surface area is

S = 4 × 135

= 540 Square feet

Because one bundle of shingles covers 27 square feet, it will take

\(\frac{540}{27}~=~20\)

Bundles of shingles to cover the rooftop of the house.

Therefore, one should buy 20 bundles of shingles to cover the rooftop.

Frequently Asked Questions on Surface Area of Pyramid

The oblique pyramid is a pyramid whose apex is not exactly above the center of the base of a pyramid.

The tetrahedron is a triangular pyramid. All the faces of the tetrahedron are triangles. In a tetrahedron, any one of its faces can be considered as the base of a pyramid and the rest three face as the lateral surfaces.