# Area of Square : Concept & Formula

## What is Square?

Its a quadrilateral in which all sides are equal and parallel

Key Features of Square
(a) All sides equal and parallel
(b) All angle measures 90 degree

## What is Area of Square?

Area of square is the amount of space covered within the boundary of square

Given above is the square ABCD of side a cm.

Observe the space inside the square (marked in blue color)

All the space inside the geometrical figure is known as Area of square

Explanation 02

Area of square can also be explained as the number of unit square present inside the given figure

Given above is the square ABCD which is made of smaller square inside it.

The side of smaller square measure 1 cm.
So the area of smaller square is 1 sq. cm

ABCD is made of 16 small squares

Hence, we can say that area of square ABCD is 16 sq. unit

## Area of Square Formula

### (A) Area Formula when side length is given

Given above is the square ABCD of side a cm

The formula for area of square is:

\mathsf{Area\ =\ Side\ \times \ Side}\\\ \\ \mathsf{Area\ =\ ( Side)^{2}}\\\ \\ \mathsf{Area\ =\ a^{2}} \\\ \\

Remember the formula as it will help to solve questions faster

### (B)Area Formula when diagonal is given

Given above is square of length a and diagonal BD of length d

Taking triangle BCD and applying Pythagoras theorem

\mathtt{BD^{2} =\ BC^{2} +\ CD^{2}}\\\ \\ \mathtt{d^{2} =\ a^{2} +\ a^{2}}\\\ \\ \mathtt{2\ a^{2} \ =\ d^{2}} \\\ \\ \mathtt{a\ =\ \frac{d}{\sqrt{2}}} \\\ \\

\mathtt{We\ know\ that:}\\\ \\ \mathtt{Area\ of\ Square\ =\ ( a)^{2}}\\\ \\ \mathtt{Putting\ the\ value\ of\ a}\\\ \\ \mathtt{Area\ of\ Square\ =\ \frac{d^{2}}{2}}

Use this formula when data of diagonal (d) is given

Example 01
Find the area of below square

Method 01
Area calculation using formula

We know that:

\mathsf{Area\ =\ ( side)^{2}}\\\ \\ \mathsf{Here,\ side\ =\ 6\ cm\ }\\\ \\ \mathsf{Area\ =\ 6^{2} \ =\ 36\ sq.\ cm\ }

Method 02
Finding number of unit squares

Unit square length = 1 cm

Unit square area = 1 sq. cm

You can see that square ABCD is made of 36 unit squares

Area of Square = Area of unit square x number of square

Area of Square = 1 x 36 = 36 sq cm

## Proof of Area of Square Formula

Given above is square ABCD with side a cm and diagonal BD

Observe that diagonal BD divides the square into two equal triangle

Area of triangle ABD
⟹ (1/2) x Base x Height

⟹ (1/2) x AD x AB

⟹ (1/2) x a x a

Similarly, Area of triangle BCD
⟹ (1/2) x BC x CD

⟹ (1/2) x a x a

Area of Square = Area of triangle ABD + BCD

Area of Square
⟹ (1/2) x a x a + (1/2) x a x a

\mathsf{\ a^{2}}

Hence Proved

## Area of Square Solved Questions

(01) Find the area of square with side 7 cm

\mathsf{We\ know\ that:}\\\ \\ \mathsf{Area\ =\ ( side)^{2}}\\\ \\ \mathsf{Here,\ side\ =\ 7\ cm\ }\\\ \\ \mathsf{Area\ =\ 7^{2} \ =\ 49\ sq.\ cm\ }

(02) Find the area of below square

Given
ABCD is a square with side 11 cm

AB = BC = CD = DA = 11 cm

\mathsf{We\ know\ that:}\\\ \\ \mathsf{Area\ =\ ( side)^{2}}\\\ \\ \mathsf{Here,\ side\ =\ 11\ cm\ }\\\ \\ \mathsf{Area\ =\ 11^{2} \ =\ 121\ sq.\ cm\ }

(03) Observe the figure carefully and find the area of square

Given above is the square ABCD

Note that the square is made of 9 unit squares

Side of smaller square = 1 cm

Area of smaller square = 1 sq. cm

Area of square ABCD
⟹ Area of small square x Number of small square

⟹ 1 x 9 = 9 sq. cm

Hence area of square ABCD is 9 sq. cm

(04) The area of square is 169 sq meter. Find the side length of square

\mathtt{We\ know\ that:}\\\ \\ \mathtt{Area\ =\ ( side)^{2}}\\\ \\ \mathtt{Here,\ Area\ =\ 169\ m\ }\\\ \\ \mathtt{169\ =\ side^{2} \ }\\\ \\ \mathtt{side\ =\sqrt{169\ } \ =\ 13}

Hence, the length of side is 13 meter

(05) The diagonal of a square is 10 cm. Find the area

\mathsf{We\ know\ the\ formula;}\\\ \\ \mathsf{Area\ of\ Square\ =\ \frac{d^{2}}{2}}\\\ \\ \mathsf{Here,\ d\ =\ 10}\\\ \\ \mathsf{Area\ =\ \frac{10^{2}}{2} \ =\frac{100}{2} \ =50} \\\ \\

Hence, 50 sq cm is the required solution