Isosceles Triangles Formulas

In geometry, an isosceles triangle is a triangle having two sides of equal length. The two angles opposite to the equal sides are equal and are always acute. Various formulas for isosceles triangles are explained below. The two important formulas for isosceles triangles are the area of a triangle and the perimeter of a triangle.

What Are the Isosceles Triangles Formulas?

An isosceles triangle has two sides of equal length and two equal sides join at the same angle to the base i.e. the third side. Thus, in an isosceles triangle, the altitude is perpendicular from the vertex which is common to the equal sides. Such special properties of the isosceles triangle help us to calculate its area as well as its altitude with the help of the isosceles triangle formulas. 

Isosceles Triangle Formulas

Area of an Isosceles Triangle: It is the space occupied by the triangle. Here we have three formulas to find the area of a triangle, based on the given parameters.

• Area = 1/2 × Base × Height

• Area = \(\frac{b}{2} \sqrt{a^{2}-\frac{b^{2}}{4}}\)
  (Here a is the equal side, and b is the base of the triangle.)

• Area = 1/2 ×abSinα
  (Here a and b are the lengths of two sides and α is the angle between these sides.)

Perimeter of an Isosceles Triangle: In an isosceles triangle, there are three sides: two equal sides and one base. In order to calculate the perimeter of an isosceles triangle, the expression 2a + b is used,

P = 2a + b

(Here, the length of the equal side is a and the length of the base is b)

Altitude of an Isosceles Triangle: In an isosceles triangle, its height is the perpendicular distance from its vertex to its base. In order to calculate the height of an isosceles triangle, the expression h = √(a²–b²/4) is used,

h = \(\sqrt{a^{2}-\frac{b^{2}}{4}}\)

Let us check a few examples to more clearly understand the use of formulas for isosceles triangles. 

Examples Using Formulas for Isosceles Triangles

Example 1: Determine the area of an isosceles triangle that has a base 'b' of 8 units and the lateral side 'a' of 5 units?

Solution:  Applying Pythagoras' theorem:

a² = (b/2)² + h²

h² = a² - (b/2)² = 5² - 4² which gives h = 3

Area 'A' = (1/2) × b × h = (1/2) 8 × 3 = 12 unit²

Answer: The area of an isosceles triangle is 12 unit².​​​​

Example 2: Find the lateral side of an isosceles triangle with an area of 20 unit² and a base of 10 units?

Solution: Using the formula of area of an isosceles triangle:

A = (1/2) b h = 20

Given b = 10,

To find: lateral side

h = 40 / 10 = 4

Applying Pythagora's theorem:

a² = (b/2)² + h² = √ ( 5² + 4²) = √41                                                                                                                                       

Answer: The lateral side of an isosceles triangle is √41.

Example 3: Calculate the area, altitude, and perimeter of an isosceles triangle if its two equal sides are of length 6 units and the third side is 8 units.

Solution:

Given a = b = 6 units, c = 8 units

To find: area, altitude, and perimeter of an isosceles triangle

Perimeter of the isosceles triangle,

P = 2×a + b

P = 2×6 + 8

= 20 units

Altitude of the isosceles triangle,

h = √(a²–b²/4)

h = √(6²–8²/4)

h = √(36−16)

h = √20 units

Area of the isosceles triangle,

A = 1/2×b×h

= 1/2×8×√20 

= √20/4 square units

Answer: