Area of an isosceles right triangle

Area of isosceles right triangle

It is the area enclosed by the three sides of the isosceles right triangle.

In the figure given below, it is the area shaded in yellow.

What is an Isosceles right triangle?

A triangle consisting of a right triangle is known as a right-angled triangle. When the two sides of this triangle other than the hypotenuse are equal, it is the isosceles right triangle.

In △ABC,

1. ∠ABC = 90°
2. AB = BC = legs = a
3. AC = hypotenuse = b
4. ∠BAC = ∠ACB = 45°

Hence it is an isosceles right triangle

The formula for finding the area

Area =\frac{1}{2} a^{2}

Where a = length of a leg

Derivation of the formula

We already know that the area of a triangle is given by the following formula

\text { Area }=\frac{1}{2} \times \text { base } \times \text { height }

In △ABC, ∠ABC = 90°

Hence BC is the base and AB is the height of the triangle.

Base = BC = a 

Height = AB = a 

\text { Area of } \triangle \mathrm{ABC}

=\frac{1}{2} \times \text { base } \times \text { height }

=\frac{1}{2} \times \mathrm{BC} \times \mathrm{AB}

=\frac{1}{2} \times \mathrm{a} \times \mathrm{a}

=\frac{1}{2} \times a^{2}

Hence it is proved that the area of an isosceles right triangle is  \frac{1}{2} a^{2}, where a is the length of the leg of the triangle.

Solved Examples.

1. In a right-angled triangle, two sides are equal, having a length of 7 cm. Find the area of this triangle?

Solution:

Since two sides of the right-angled triangle are equal, it is an isosceles right triangle.

Length of the leg = a = 7 cm

\text { Area of the right- angled triangle }

=\frac{1}{2} \times a^{2}

=\left(\frac{1}{2} \times 7^{2}\right) \mathrm{cm}^{2}

=\left(\frac{1}{2} \times 49\right) \mathrm{cm}^{2}

=24.5 \mathrm{~cm}^{2}

 The area of this triangle is 24.5 \mathrm{~cm}^{2}

2. The area of a triangle is 18 cm². What is the length of the hypotenuse if it is an isosceles right triangle?

Solution:

We know that 

\text { Area }=\frac{1}{2} \times a^{2}

Where a = length of the leg

It is given that

\text { Area }=18 \mathrm{~cm}^{2}

\Rightarrow \frac{1}{2} \times a^{2}=18

\Rightarrow a^{2}=18 \times 2=36

\Rightarrow a=\sqrt{36}=6 \mathrm{~cm}

Since two sides are equal in an isosceles right triangle, according to Pythagoras theorem we can write that

{\text{Side}} ^{2}+{\text{Side}} ^{2}={\text{hypotenuse}} ^{2}

{\text{Hypotenuse}} ^{2}=a^{2}+a^{2}=2 a^{2}=2 \times 6^{2}=2 \times 36=72

{\text{Hypotenuse}} =\sqrt{72}=6 \sqrt{2} \mathrm{~cm}

Hence the length of the hypotenuse is 6√2 cm.