Isosceles Triangle (original) (raw)

Last Updated : 23 Jul, 2025

An **isosceles triangle is a type of triangle in geometry that has at **leasttwo sides of equal length. The angles opposite these sides are also equal.
Suppose we have any triangle △PQR then it is an **isosceles triangle if any one of the given conditions is true:

In the figure below the Side **AC = BC, also **∠A = ∠B making it an **Isosceles Triangle.

Isosceles Triangle

**Note: **Every equilateral triangle is an isosceles triangle .

Real-life Examples of Isosceles triangle:

Examples of Isosceles Triangle

**Read More: Real-Life Examples of Isosceles triangle

Angles of Isosceles Triangle

A triangle has three angles so an isosceles triangle also has three angles but an isosceles triangle is a special case as it has two angles of the three angles equal. The angle sum property of the triangle also holds for the Isosceles triangle. Suppose we have an isosceles triangle △ABC where AB = AC, and ∠B = ∠C. If the unknown angle ∠A is given then we can easily find the other angle of the Isosceles triangle. For example,

**Example: In Isosceles triangle △ABC where ∠B = ∠C and ∠A = 80°. Find other angles.
**Solution:

We know that, in any triangle △ABC

∠A + ∠B + ∠C = 180°

Also, ∠B = ∠C and ∠A = 80°

Using angle sum property of triangle,

80° + ∠B + ∠B = 180°

⇒ 2∠B = 100°
⇒ ∠B = 50°

Thus, measure of other two angles of an isosceles triangle is 50°.

Types of Isosceles Triangles

Isosceles triangles are classified into three types depending on the measures of angles, which include,

Examples-of-Triangles-1

Types of Isosceles Triangle

**Isosceles Right Triangle

An isosceles triangle that has a right angle is called an Isosceles Right triangle. Examples of isosceles right triangles are,

**Isosceles Obtuse Triangle

An isosceles triangle in which any one angle is obtuse angles and the other two are acute angles is called an Isosceles Acute triangle. Some examples of isosceles obtuse triangles are,

**Isosceles Acute Triangle

An isosceles triangle in which all the angles are acute is called an Isosceles Acute triangle. Some examples of isosceles acute triangles are,

Properties of Isosceles Triangles

The following are some important characteristics of an isosceles triangle:

Isosceles Triangle Theorem

There are two common theorems related to isosceles triangles i.e.,

Base Angle Theorem

Angle Theorem states that,

****"If two sides in any isosceles triangle are equal then the angle opposite to them are also equal."**

Isosceles Triangle Theorem

The converse of the base angle theorem is also true which states that

****"If two angles in any isosceles triangle are equal then the side opposite to them are also equal."**

If we have an Isosceles triangle ABC then,

**AB = AC ⟺ ∠ABC = ∠ACB

Isosceles Triangles Formulas

The height, perimeter, and area are the three basic formulas of an isosceles triangle, which are discussed below.

Perimeter of Isosceles Triangle

The perimeter of an isosceles triangle is equal to the sum of its three side lengths. As an isosceles triangle has two equal sides, the perimeter of the isosceles triangle will be (2a + b) units, where "a" is the length of the two equal sides and "b" is the base length.

Perimeter of an Isosceles Triangle

**Perimeter of an Isosceles Triangle = (2a + b) units

Where,

**Learn more: **Perimeter of a Triangle

Isosceles Triangle Area

The total region bounded by the three sides of a triangle in a two-dimensional plane is known as the area of a triangle. The area of an isosceles triangle is equal to half the product of its base length and its height.

Area of Isosceles Triangle

**Area of an Isosceles Triangle = ½ × base × height

Area of an Isosceles Triangle = \frac{1}{2}\times b\times\sqrt{[a^{2}-(b/2)^{2}]}

Where,

Isosceles Triangle Altitude

The perpendicular drawn from the apex angle bisects the base of the isosceles triangle and the apex angle. The formula to calculate the height of an isosceles triangle if its side lengths are given is as follows:

Height of an Isosceles Triangle (h) =\sqrt{[a^{2}-(b/2)^{2}]}

Where,

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Solved Examples on Isosceles Triangle

**Example 1: Determine the perimeter of an isosceles triangle with equal sides measuring 7 cm and a base length of 10 cm.
**Solution:

Given,
Lengths of equal sides of the triangle (a) = 7 cm
Base length (b) = 10 cm

We know that,
Perimeter of an isosceles triangle (P) = 2a + b
⇒ P = 2 × 7 + 10
⇒ P = 14 + 10 = 24 cm

Thus, the perimeter of the given isosceles triangle is 24 cm.

**Example 2: Determine the area of an isosceles triangle whose base length is 14 cm, and its height is 7.5 cm.
**Solution:

Given,
Height (h) = 7.5 cm
Base length (b) = 14 cm

We know that, Area of an isosceles triangle (A) = ½ × b × h
⇒ A = ½ × 14 × 7.5
⇒ A = ½ × 105
⇒ A = 52.5 sq. cm

Hence, the area of the given isosceles triangle is 52.5 sq. cm.

**Example 3: Determine the height of an isosceles triangle with equal sides measuring 13 cm and a base length of 10 cm.
**Solution:

Given,
Lengths of equal sides of the triangle (a) = 13 cm
Base length (b) = 10 cm

We know that,

Height of an isosceles triangle (h) = \sqrt{[a^{2}-(b/2)^{2}]}
⇒ h = \sqrt{[13^{2}-(10/2)^{2}]}
⇒ h = \sqrt{[13^{2}-5{}^{2}]}
⇒ h = \sqrt{[169-25]}
⇒ h = √144 = 12 cm

Hence, the height of the given isosceles triangle is 12 cm.