Isosceles Triangle

An Isosceles triangle is defined as a triangle that has two equal sides. We know that a triangle has three angles and if any two of them are equal we call this triangle an Isosceles triangle as well. Thus, we can say that if a triangle has two sides equal and two angles equal then it is an isosceles triangle.

Triangle is divided into three categories on the basis of equality of side length.

• Scalene triangle (All three sides are unequal)
• Isosceles triangle (Only two sides are equal)
• Equilateral triangle (All three sides are equal)

In this article, we will learn about the isosceles triangle area, its parameter, properties, examples, and others in detail.

What is an Isosceles Triangle?

An isosceles triangle is a triangle with two equal sides and two equal angles. In an isosceles triangle, the two sides of equal length are called the legs, and the third side of the triangle is called the base. An equilateral triangle is also an isosceles triangle, but the contrary is not always true. In the figure given below, ∆ABC is an isosceles triangle, where the sides AC and BC are equal. We know that the angles opposite the equal sides are equal. So, ∠A and ∠B are equal.

Isosceles Triangle Definition

Any triangle with two equal sides is called an Isosceles triangle. An isosceles triangle also has two equal angles as in a triangle equal sides have equal angles opposite to it.

Suppose we have any triangle PQR then it is any isosceles triangle if any one of the given conditions is true.

• PR = QR
• ∠P = ∠Q

Examples of Isosceles Triangle

In real life we can see the examples of Isosceles triangle, some of these examples are illustrated in the following image.

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, the measure of the 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,

• Isosceles Acute Triangle
• Isosceles Right Triangle
• Isosceles Obtuse Triangle

Let’s learn about them in detail in this article,

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,

• Triangle with angles 50°, 50° and 80°
• Triangle with angles 65°, 65° and 50°, etc.

Isosceles Right Triangle

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

• Triangle with angles 45°, 45° and 90°

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,

• Triangle with angles 40°, 40° and 100°
• Triangle with angles 35°, 35° and 110°, etc.

Properties of Isosceles Triangles

The following are some important characteristics of an isosceles triangle:

• An isosceles triangle will always have at least two equal sides and two equal angles.
• In an isosceles triangle, the two sides of equal length are called the legs, and the third side of the triangle is called the base.
• The angle between the legs of an isosceles triangle is called the apex angle or vertex angle.
• The perpendicular drawn from the apex angle bisects the base of the isosceles triangle and the apex angle.
• The perpendicular drawn from the apex angle is also known as the line of symmetry as it divides the isosceles triangle into two congruent triangles.

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that,

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

The converse of this 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 = ∠ADC

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 = (2a + b) units

Where 

• “a” is the length of the two equal sides, and
• “b” is the base length.

Learn more about, 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 an Isosceles Triangle = ½ × base × height

Learn more about, Area of Isosceles Triangle

• If the three side lengths of an isosceles triangle are given, then its area can be calculated using Heron’s formula.

Area of an Isosceles Triangle = 

Where 

• “a” is the length of the two equal sides, and
• “b” is the base length.

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) =

Where 

• “a” is the length of the two equal sides, and
• “b” is the base length.

Related Resources:

• Equilateral Triangle
• Heron’s Formula
  
• Area of Isosceles Triangle

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) = 

⇒ 

⇒ 

⇒ 

⇒ h = √144 = 12 cm

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

Example 4: Determine the area of an isosceles triangle with equal sides measuring 25 cm and a base length of 14 cm.

Solution:

Given data:

Lengths of equal sides of the triangle (a) = 7 cm

Base length (b) = 10 cm

We know that,

Area of an Isosceles Triangle = 

⇒ 

⇒ 

⇒ 

⇒ A = 7 × √576 = 7 × 24

⇒ A = 168 sq. cm

Hence, the area of the given isosceles triangle is 168 sq. cm.

Practice Questions on Isosceles Triangle

Q1: What is the perimeter of a field whose shape is in form of a triangle having two equal sides of 10m and third side being 5m.

Q2: If the height of a isoceles triangle is 4 cm and base is 6 cm find the length of equal sides.

Q3: If the equal angles of an isosceles triangle measures 60°, find the third angle. Also state what type of the triangle is this based on the value of all the three angles.

Q4: If the perimeter of an isosceles triangle is 25 cm and the unequal side is 5 cm. Find the length of the two equal sides.

Q5: Find the area of an isosceles triangle whose base is 12 cm and equal sides are 13 cm.

FAQs on Isosceles Triangles

1. What is Isosceles Triangle?

An isosceles triangle is a triangle with at least two sides equal and the corresponding angles opposite to them are also equal.

2. What is the Perimeter of an Isosceles Triangle?

The perimeter of an isosceles triangle is equal to the sum of its three side lengths. If the isosceles triangle has equal side a and unequal side b then,

Perimeter = a + a + b

Perimeter = 2a + b

3. What is the Area of the Isosceles Triangle?

The area of an isosceles triangle is the space occupied by the boundary of the isosceles triangle. If the base of the isosceles triangle is b units and its height is h units then its, area is given as,

A = (½)×bh  units²

4. What are the Different Types of Isosceles Triangles?

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

•  Isosceles Acute triangle 
• Isosceles Right triangle
• Isosceles Obtuse triangle

5. What is an Isosceles Right Triangle?

Isosceles right triangle, is a triangle which is a isosceles triangle and has a right angle. Here, one interior angle of the triangle is equal to 90°, while the other angles are acute angles and each measures 45°.

6. What are the properties of the Isosceles Triangle?

Some important properties of the isosceles triangle are,

• They have two sides of equal length.
• Angles opposite to the equal sides are equal in measure.
• The line of symmetry of the isosceles triangle is the perpendicular drawn from the vertex of the Isosceles triangle.