Area of Isosceles Triangle

An isosceles triangle is a special type of triangle where two sides are of same length, and the angles opposite them are also equal. This unique property allows us to use specific formulas to calculate its area efficiently.

1.0What is an Isosceles Triangle?

An isosceles triangle is defined by having at least two sides of equal length. The angles opposite these equal sides are also equal. This type of triangle features:

• Two equal sides (legs) and a base.
• Two equal angles (base angles) opposite the equal sides.
• Symmetry along the line that bisects the vertex angle.

2.0Area of Isosceles Triangle Formula

To find the area of an isosceles triangle, we can use several formulas based on the information provided:

Standard Formula:

If the base b and the height h (the vertical distance from the base to the opposite vertex) are known, the area A can be computed as:

$A=\frac{1}{2}\times b\times h$

3.0Area of Isosceles Triangle Without Height

If the height is not given, you can still find the area using other methods, such as:

1. Using Sides:

Suppose you know the lengths of both equal sides a and the base b. You can use the Pythagorean theorem to find the height h:

$h=\sqrt{a^2-{\left(\frac{b}{2}\right)}^2}$

Then, substitute h into the standard formula to find the area.

So, the area of isosceles triangle

$=\frac{1}{2}\times \mathrm{b}\times \mathrm{h}$

$=\frac{1}{2}\times \mathrm{b}\times \sqrt{{\mathrm{a}}^2-{\left(\frac{\mathrm{b}}{2}\right)}^2}$

$=\frac{1}{2}\times b\times \sqrt{a^2-{\left(\frac{b}{2}\right)}^2}$

2. Using Trigonometry:

If the angle θ between the two equal sides is known, the area can be found using:

$A=\frac{1}{2}\times a^2\times \sin (\theta )$

4.0Area of Isosceles Triangle with Sides 

When you know the lengths of all three sides, a, a, and b, you can use Heron's Formula to find the area. First, calculate the semi-perimeter s:

$S=\frac{a+a+b}{2}$

$S=\frac{2a+b}{2}$

$S=a+\frac{b}{2}$

Then, apply Heron's formula:

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{s(s-a)(s-a)(s-b)}$

$A=\sqrt{s(s-a{)}^2(s-b)}$

$A=(s-a)\sqrt{s(s-b)}$

$A=\left(a+\frac{b}{2}-a\right)\sqrt{\left(a+\frac{b}{2}\right)\left(a+\frac{b}{2}-b\right)}$

$A=\frac{b}{2}sqrt\left(a+\frac{b}{2}\right)\left(a-\frac{b}{2}\right)=\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}$

Area of isosceles triangle with sides using heron’s formula is =$\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}$ .

5.0Area of Isosceles Right Triangle

For an isosceles right triangle, where the two equal sides are perpendicular to each other, the area can be computed directly as:

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

where a is the length of each of the equal sides.

6.0Area of Isosceles Triangle Solved Examples

Example 1: Find the area of an isosceles triangle with base 10 units and height 6 units.

Solution: 

b = 10 units, h = 6 units, A = ?

Area of isosceles triangle:

$=\frac{1}{2}\times b\times h$

$=\frac{1}{2}\times 10\times 6$

= 30 square units

Example 2: Find the area of an isosceles triangle with equal sides of length 5 units and base 6 units.

Solution: 

a = 5 units, b = 6 units, A = ?

Area of isosceles triangle:

$=\frac{1}{2}\times \mathrm{b}\times \mathrm{h}$

$=\frac{1}{2}\times 6\times 4$

= 12 square units

Example 3: Calculate the area of the isosceles right tringle with legs of 7 units each.

Solution: 

a = 7 units , A = ?

Using formula of the area of an isosceles triangle

$\mathrm{A}=\frac{1}{2}\times {\mathrm{a}}^2=\frac{1}{2}\times 7^2=\frac{1}{2}\times 49=\frac{49}{2}$

A = 24.5 square units

Example 4: Determine the area of an isosceles triangle where the equal sides are 8 units long and the included angle between them is 60°.

Solution: 

a = 8 units. θ = 60°

Using the trigonometric area formula.

⇒ $A=\frac{1}{2}\times a^2\times \sin \theta$

⇒ $A=\frac{1}{2}\times 6^2\times \sin 60^{\circ }$

⇒ $A=\frac{1}{2}\times 64\times \frac{\sqrt{3}}{2}$

⇒ $A=16\sqrt{3}$ square units

Example 5: Find the area of an isosceles triangle with sides 13 units, 13 units and a base of 10 units Heron’s formula.

Solution: 

a = 13 units, b = 13 units, c = 10 units

Using Heron’s formula

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$S=\frac{a+b+c}{2}$

$S=\frac{13+13+10}{2}$

S = 18 units

Using Heron’s formula

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{18(18-13)(18-13)(18-10)}$

$A=\sqrt{18\times 5\times 5\times 8}=\sqrt{3600}$

A = 60 square units

7.0Area of Isosceles Triangle Practice Questions

1. Calculate the area of an isosceles triangle with sides 7 units, 7 units, and a base of 10 units.
2. Find the area of an isosceles right triangle with legs of 8 units.
3. Find the area of an isosceles triangle where the equal sides are 15 units, and the base is 18 units.
4. Determine the area of an isosceles right triangle where each leg is 9 units long.
5. A triangle has two equal sides of 6 units and an included angle of 45°. Calculate its area.
6. Using Heron's formula, find the area of an isosceles triangle with sides 7 units, 7 units, and 12 units.

8.0Sample Questions on Area of Isosceles Triangle

1. What is the formula to calculate the area of an isosceles triangle?

Ans: The most common formula is $A=\frac{1}{2}\times \text{ base }\times \text{ height }$, where the base is the length of the unequal side, and the height is the perpendicular distance from vertex to the base.

2. Can I use Heron's formula to find the area of an isosceles triangle?

Ans: Yes, Heron's formula can be used when all three sides of the isosceles triangle are known. First, calculate the semi-perimeter s, then apply the formula $A=\sqrt{s(s-a)(s-b)(s-c)}$ .

3. What if I only know the lengths of the two equal sides and the angle between them?

Ans: If you know the lengths of the two equal sides and the included angle, you can use the trigonometric formula $A=\frac{1}{2}\times a^2\times \sin (\theta )$ , where a is the length of the equal sides, and θ is the included angle.

4. How do I calculate the height of an isosceles triangle if only the side lengths are known?

Ans: The height h can be determined using the Pythagorean theorem: $h=\sqrt{a^2-{\left(\frac{b}{2}\right)}^2}$ , where a represents the length of the equal sides, and b represents the base.