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Area of Isosceles Triangle Formula

The word isosceles triangle is a type of triangle, it is the triangle that has two sides the same length. If all three sides are equal in length then it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle. In this article, we will discuss the isosceles triangle and area of isosceles triangle formula. Let us begin learning!

Area of Isosceles Triangle Formula

Definition of Isosceles Triangle:

An isosceles triangle is a triangle with two sides of equal length and two equal internal angles adjacent to each equal sides. It is unlike an equilateral triangle where we can use any vertex to find out the altitude. Thus in an isosceles triangle, we have to draw a perpendicular from the vertex which is common to the equal sides.

Therefore, in an isosceles triangle, two equal sides join at the same angle to the base i.e. the third side. These special properties of the isosceles triangle will help us to calculate its area from just a couple of pieces of information.

Let us learn the methods to find out the area, altitude, and perimeter of such an isosceles triangle.

Properties:

  • The unequal side of an isosceles triangle is normally referred to as the ‘base’ of the triangle.
  • The base angles of the isosceles triangle are always equal.
  • If the 3rd angle is a right angle, it is called a “right isosceles triangle”.
  • The altitude of a triangle is a perpendicular distance from the base to the topmost

Procedure to compute the area of an isosceles triangle:

Step-1: Find the isosceles triangle’s base. The base is the easy part, and just use the third unequal side as the base. The length of base will be b.

Step-2: Draw a perpendicular line between the base to the opposite vertex.  The length of this line will be the height of the triangle, so label it as h. After computing h we can find the area. In the isosceles triangle, this line will always hit the base at its exact midpoint.

Step-3: The perpendicular line will divide the triangle into two equal right-angled triangles. The hypotenuse s of the right triangle is one of the two equal sides of the isosceles. Its base will be half of the base i.e. b/2. Thus using the Pythagorean Theorem we will determine the length of perpendicular i.e. h using,

h= \(\sqrt{s^2-(\frac{b}{2})^2}\)

Where,

S The hypotenuse of a right-angled triangle
B Length of base
H Height of triangle

Step-4: Put the base and height into the area formula. The formula is as follows:

A = \(\frac{b X h}{2}\)

A Area of the isosceles triangle
B Length of base
H Height of triangle

Remember to write the answer in terms of square units.

Solved Examples

Q.1: Find the area, altitude, and perimeter of an isosceles triangle given a = 5 cm, b = 9 cm, c =5 cm?

Solution:

Given,

a = 5 cm

b = 9 cm

C= 5 cm.

Perimeter of an isosceles triangle

= a + b + c cm

= 5 + 9 + 5 cm

= 19 cm

Height of an isosceles triangle

h = \(\sqrt{s^2-(\frac{b}{2})^2}\)

= \(\sqrt{5^2-(\frac{9}{2})^2}\)

= \(\sqrt{25–\frac{81}{4}}\)

= 2.18 cm

Area of an isosceles triangle,

A = \(\frac{b X h}{2}\)

= \(\frac{9 X 2.18}{2}\)  cm²

= 9.81 cm²

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