An isosceles triangle’s area is the amount of space it encloses in two dimensions. The general formula for the area of a **triangle** is half the product of its base and height. To help you grasp this idea better, we’ve included a full description of the isosceles triangle area, its formula, and derivation, as well as a few solved sample questions.

Table of Contents

- 1 What is Isosceles Triangle?
- 2 What is the Area of Isosceles Triangle?
- 3 List of Formulas to Find Isosceles Triangle Area
- 4 How to Find the Area of an Isosceles Triangle?
- 5 Derivation for Isosceles Triangle Area Using Heron’s Formula
- 6 Area of Isosceles Right Triangle Formula
- 7 Area of Isosceles Triangle Using Trigonometry
- 8 Solved Example
- 9 FAQs

## What is Isosceles Triangle?

Triangles are three-sided enclosed polygons that can be classified as equilateral, isosceles, or scalene depending on the length of their sides. Isosceles triangles are those with at least two sides of identical length.

An isosceles triangle has specific qualities that differentiate it from other forms of triangles, which are given below:

- There are two equal sides and
**angles**. - An isosceles triangle’s legs are its two equal sides, and the angle between them is known as the vertex angle or apex angle.
- The side opposite the vertex angle is known as the base, and base angles are equal.
- The base and apex angles are bisected from the apex angle’s
**perpendicular**. - The isosceles triangle is divided into two congruent triangles by the perpendicular drawn from the apex angle, which is also known as the line of symmetry.

## What is the Area of Isosceles Triangle?

The area of an isosceles triangle is the total amount of space covered inside its boundaries. If you know the height and base of an isosceles triangle, you can easily compute its area. The area of an isosceles triangle is calculated by multiplying half of its base and height.

Area = (1/2) × (base) × (height)

where “base” refers to the isosceles triangle’s base length and “height” refers to the perpendicular distance between the base and the opposing vertex.

The area of an isosceles triangle is measured in square units, such as square centimetres, square inches, or square metres, depending on the base and height of the triangle.

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Also,

The perimeter of the isosceles triangle | P = 2a + b |

The altitude of the isosceles triangle | h = √(a^{2} − b^{2}/4) |

## List of Formulas to Find Isosceles Triangle Area

Below is the list of the formulas used to find different Isosceles Triangle Area:

Formulas to Find Area of Isosceles Triangle | |

Using base and Height | A = ½ × b × hwhere b = base and h = height |

Using all three sides | A = ½[√(a^{2} − b^{2} ⁄4) × b]a is the measure of equal sidesb is the base of triangle |

Using the length of 2 sides and an angle between them | A = ½ × a × b × sin(α)a is the measure of equal sidesb is the base of triangle |

Using two angles and length between them | A = [a^{2}×sin(β)×sin(α)/ 2×sin(2π−α−β)]a is the measure of equal sidesb is the base of triangleα is the measure of equal anglesβ is the angle opposite to the base |

Area formula for an isosceles right triangle | A = ½ × a^{2}a is the measure of equal sides |

## How to Find the Area of an Isosceles Triangle?

To calculate the area of an isosceles triangle, follow these steps.

Step 1: Determine the length (l) and breadth (b) of the given triangle.

Step 2: Multiply the values from step 1 and divide by 2.

Step 3: The value produced is the needed area, measured in m^{2}.

## Derivation for Isosceles Triangle Area Using Heron’s Formula

The area of an isosceles triangle can be calculated using Heron’s formula, as explained below.

According to Heron’s formula,

**Area = √[s(s−a)(s−b)(s−c)]**

Where, s = ½(a + b + c)

Now, for an isosceles triangle,

s = ½(a + a + b)

= s = ½(2a + b)

Or, s = a + (b/2)

Now,

Area = √[s(s−a)(s−b)(s−c)]

Or, Area = √[s (s−a)^{2} (s−b)]

= Area = (s−a) × √[s (s−b)]

Substituting the value of “s”

= Area = (a + b/2 − a) × √[(a + b/2) × ((a + b/2) − b)]

= Area = b/2 × √[(a + b/2) × (a − b/2)]

**Or, area of isosceles triangle = b/2 × √(a**^{2}** − b**^{2}**/4)**

## Area of Isosceles Right Triangle Formula

The formula for Isosceles Right Triangle Area = ½ × a^{2} |

**Deriviation:**

An isosceles right triangle’s perimeter is equal to the sum of its sides.

Assume the two sides are equal. Using Pythagoras’ theorem, the uneven side is calculated to be a√2.

Therefore, the perimeter of an isosceles right triangle is a+a+a√2.

= 2a+a√2

= a(2+√2)

= a(2+√2)

## Area of Isosceles Triangle Using Trigonometry

Using Length of 2 Sides and the angle between them,

A = ½ × b × c × sin(α)

Using 2 Angles and Length between them

A = [c^{2}×sin(β)×sin(α)/ 2×sin(2π−α−β)]

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## Solved Example

- Find the area of an isosceles triangle with a base length of 10 cm and a height of 17 cm.

Solution,

Base of the triangle (b) = 10 cm

Height of the triangle (h) = 17 cm

Area of Isosceles Triangle = (1/2) × b × h

= (1/2) × 10 × 17

= 5 × 17

= 85 cm^{2}

The area of the given isosceles triangle is 85 cm^{2}.

- Find the length of the base of an isosceles triangle with an area of 243 cm
^{2}and an altitude of 9 cm.

Solution,

Area of the triangle, A = 243 cm^{2}

Height of the triangle (h) = 9 cm

The base of the triangle = b =?

Area of Isosceles Triangle = (1/2) × b × h

243 = (1/2) × b × 9

243 = (b × 9)/2

b = (243 × 2)/9

b = 54 cm

The altitude of the given isosceles triangle is 54 cm.

- Find the lengths of the equal sides of an isosceles triangle with a base of 24 cm and an area of 60 cm
^{2}.

Solution,

We know:

The base of the isosceles triangle = 24 cm

Area of the isosceles triangle = 60 cm^{2}

Area of isosceles triangle = b/2 × √(a^{2} − b^{2}/4)

Therefore,

60 = (24/2)√(a^{2} − 24^{2}/4)

60 = 12√(a^{2} − 144)

5 = √(a^{2}−144)

Squaring both sides, we get,

25 = a^{2}−144

a^{2} = 169

= a = 13 cm

The length of the equal sides of the given isosceles triangle is 13 cm.

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## FAQs

**What is the formula for the area of an isosceles triangle?**

To compute the area of an isosceles triangle, use the following formula:

A = ½ × b × h

**What is the formula for the perimeter of an isosceles triangle?**

The perimeter of an isosceles triangle can be calculated using the following formula:

P = 2a + b

**What is the area of an isosceles triangle?**

The area of a figure is defined as the space enclosed by its bounds. As a result, the area of an isosceles triangle is defined as the space occupied by the triangle itself.

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