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