Perimeter of Isosceles Triangle: Definition, Formula, Types & Solved Examples

Isosceles Triangle is the type of triangle in which any two of the sides are equal. The angles that are found the opposite to these equal sides are also equal. We can calculate the area of an isosceles triangle with the help of the length of its sides. For triangle ABC whose sides AB and AC are equal, so, ∠ B will also be equal to ∠ C. The theorem that explains the definition of the isosceles triangle is that “if the two sides of a triangle are congruent, then the angle opposite to these sides are congruent”. The perimeter of an isosceles triangle refers to the total length of its boundary which is the sum of all its sides.

Key Words: Isosceles Triangle, Triangles, Perimeter, Angles, Base, Height, Congruency, Hypotenuse, Vertex, Right Isosceles Triangle, Obtuse Isosceles Triangle, Acute Isosceles Triangle

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Properties of An Isosceles Triangle

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An isosceles triangle is a type of a triangle along with equilateral triangle and scalene triangle. It can be defined as a triangle with two equal sides. Here are some of the important properties of an isosceles triangle:

• Any of the two sides are equal in an isosceles triangle, and the unequal side is known as the triangle’s base.
• The angles that are opposite to the two equal sides, will also be equal.
• An Isosceles triangle’s height is taken from the base of the triangle to the vertex of the triangle.
• The remaining unequal angle of a right isosceles triangle is equal to 90 degrees.
• An isosceles triangle can be generally classified into different types, such as

the Isosceles acute triangle, the Isosceles right triangle and the Isosceles obtuse triangle.

Isosceles Triangle

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What is an Isosceles Acute Triangle?

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As we are aware of the fact that the various dimensions of a triangle are base, legs, and height. The axis of symmetry of an isosceles triangle is found along its base’s perpendicular bisector. An Isosceles triangle can be classified as either acute, right or obtuse depending upon the angle between the two legs. The isosceles triangle is known as an Isosceles acute triangle if the angles opposite to the equal legs are equal and are lower than 90 degrees.

Isosceles Acute Triangle

Read More: Properties of Triangle

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What is an Isosceles Right Triangle?

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A right isosceles triangle consists of two equal sides, in which one equal side is the perpendicular and another one is the base of the triangle. The third unequal side is called the hypotenuse. Therefore, the Pythagoras theorem can be applied here, where the square of the hypotenuse is equal to the sum of the square of base and perpendicular. Let’s assume that the sides of the right isosceles triangle are a, a, and h, where a is the length of two equal sides and h is the length of the hypotenuse, then; h = √(a²+a²) = √2a²= a√2 or h = √2a

Isosceles Right Triangle

Read More: Right Angle Triangle Theorem

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What is an Isosceles Obtuse Triangle?

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An obtuse triangle is a triangle where one angle is larger than 90 degrees. Also, it is impossible to draw a triangle consisting of more than two obtuse angles. The obtuse triangle can be of two main types, that are, scalene triangles or isosceles triangles. Therefore, the isosceles obtuse triangle is a triangle consisting of two equal sides with an obtuse angle.

Isosceles Obtuse Triangle

Read More: Congruence of Triangles

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Perimeter of an Isosceles Triangle

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The perimeter of any kind of shape is calculated by measuring the total boundary of that shape. In the same way, the perimeter of an isosceles triangle is measured as the sum of all three sides of an isosceles triangle. The perimeter of an isosceles triangle can be measured if its base and side are available. The formula for the perimeter of an isosceles triangle is given by:

The perimeter of an isosceles triangle’s formula is as follows:

P = 2a + b Units

Here ‘a’ is the two equal sides’ length of an isosceles triangle and ‘b’ is the base of that triangle.

Example: Calculate the perimeter of an isosceles triangle whose equal sides are 10 cm and the base is given as 5 cm.

Solution: To calculate the perimeter of an isosceles triangle, we will use the perimeter formula:

P = 2a + b Units

Here, the length of equal sides is 10 cm and the base is given as 5 cm. So,

Perimeter of the isosceles triangle = 2 x 10 + 5 = 20 + 5 = 25 cm.

Thus, the perimeter of the given isosceles triangle is 25 cm.

Read More: The Perimeter of a Triangle

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Things To Remember

• Isosceles Triangle can be defined as a type of triangle that has two equal sides and the angles that are found the opposite to these equal sides are also equal. The third unequal side is called the hypotenuse.
• An Isosceles triangle can be classified as either acute, right or obtuse depending upon the angle between the two legs.
• The isosceles triangle is known as an Isosceles acute triangle if the angles opposite to the equal legs are equal and are lower than 90 degrees.
• A right isosceles triangle consists of two equal sides, in which one equal side is the perpendicular and another one is the base of the triangle.
• An obtuse triangle is a triangle where one angle is larger than 90 degrees.
• The perimeter of an isosceles triangle refers to the sum total of all the sides of an isosceles triangle.
• The formula for the perimeter of an isosceles triangle is P = 2a + b units. Here ‘a’ is the length of the two equal sides of an isosceles triangle and ‘b’ is the base of the triangle.

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