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Surface Area of a Square Pyramid

A pyramid has one polygonal surface called base which is connected with triangular-shaped side faces. It is a three-dimensional object obtained by connecting its base to an apex. Based on the shape of the base, the type of pyramid is named. If the base of the pyramid is a square, then it is called a square pyramid. We are going to learn about the square pyramid and their surface areas on this page.

After the completion of this article, you will be able to learn the following topics. 

1) Definition of a square pyramid and surface area.
2) Formula and method to calculate the surface area of a square pyramid.
3) Examples.

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Definition of Square Pyramid and Surface Area

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If the base of a pyramid is Square, then it is called Square Pyramid. It consists of one square-shaped base and four triangular lateral faces. A figure of a square pyramid is as shown.
 
Surface Area of a Square Pyramid

In this figure, let the side length of the base be $a$ and slant height be $s$.

Note that the slant height is the length of perpendicular drawn from the apex to the side of any triangular lateral face.

The surface area of a pyramid is defined as the sum of all its surfaces. It includes the sum of the area of base and area of all the triangular lateral faces. Also, the area of all the side surfaces is known as lateral surface area.

Formula to calculate Surface Area of a Square Pyramid

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As we discussed that a square pyramid is made up of a square base and four triangular lateral faces connected to the base. The sum of the area of the square base and area of the four triangular lateral faces gives the surface area. The sum of the area of lateral faces is called lateral surface area, in short LSA and the sum of the lateral surface area and area of the base is termed as total surface area, i.e. TSA or simply surface area.

So, Surface Area = Base Area + Lateral Area

For the above figure, the surface area is calculated as follows:

Base Area = $a^2$

Lateral Area= $\frac{1}{2}\ as\ +\ \frac{1}{2}\ as\ +\ \frac{1}{2}\ as\ +\ \frac{1}{2} as$ = $2as$

Surface Area = Base Area + Lateral Area

                  $SA$ =  $a^2\ +\ 2as$      where a-base side length and s-slant height

We may also write the formula for surface area as under:

Total surface area = Lateral surface area + area of base

And Lateral surface area of square pyramid can be directly calculated by the formula:

LSA = $\frac{1}{2}$ perimeter of base $\times$ slant height

= $\frac{1}{2}$ $\times$ 4a $\times$ s = 2as

So, TSA = a$^{2}$ + 2as

Examples

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Example 1: Find the surface area for the following square pyramid:

Surface Area of a Square Pyramid Examples
 
Solution:

Base Area = $7\ \times\ 7$ = $49$ sq units

Lateral Area = $2\ \times\ 7\ \times\ 12$ = $168$ sq units

Surface Area = $49\ +\ 168$ = $217$ sq units
Examples 2: Find the surface area of a square pyramid with a side length of base $5\ cm$ and slant height is $3\ cm$.

Solution:

Lateral surface area = $2\ \times\ 5\ \times\ 3$ = $30$ sq cm

Area of base = $5^{2}$ = $25$ sq cm

Surface Area is given by

$SA$ = LSA + area of base

$SA$ = $25\ +\ 30$

$SA$ = $55$ sq cm
Exmaple 3: The figure of a square pyramid is as shown. Find the Surface Area.

Examples on Surface Area of a Square Pyramid
 
Solution:

Formula to find Surface Area is  $SA$ = $a^2\ +\ 2as$

From the above figure, $a$ = $16$ and pyramid height $(h)$ = $15$.

Apply Pythagoras theorem to find slant height $(s)$

$s^2$ = $15^2\ +\ 8^2$

$s^2$ = $225\ +\ 64$

$s^2$ = $289$

$s$ = $\sqrt{289}$

$s$ = $17$

$SA$ = $16\ \times\ 16\ +\ 2\ \times\ 16\ \times 17$

$SA$ = $256\ +\ 544$

$SA$ = $800$ sq units
Example 4: The surface area of a square pyramid is $360\ sq\ m$ and slant height is $13\ m$. Find the area of the base.

Solution:

Given, $SA$ = $360$ and $s$ = $13$

Formula is $SA$ = $a^2\ +\ 2\ as$

$360$ = $a^2\ +\ 2a(13)$

$360$ = $a^2\ +\ 26a$

$a^2\ +\ 26a\ -\ 360$ = $0$

$(a\ +\ 36)(a\ -\ 10)$ = $0$

$a$ = $-36$,$a$ = $10$

Since lengths cannot be negative

$a$ = $10$

The side length of base is $10\ m$

The base area is $10\ \times\ 10$ = $100\ sq\ m$.
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