Area of Isosceles Triangle - Derivation, Different Formulas based on Conditions and Area of a triangle with all three sides

What is an Isosceles Triangle?

Isosceles is derived from the Greek words iso, which means same and Skelos, which means legs. Putting both together, we get a term that means same legged figure. Therefore, an isosceles triangle is one with at least two sides (legs) of the same length. This characteristic corresponds to two angles of the same triangle. There are two identical sides of an isosceles triangle and two similar angles. The angles opposite to the identical sides are equal in an isosceles triangle. We can obtain two right-angled triangles if we project a perpendicular line from the point of convergence of two identical sides to the base. This perpendicular line divides the base into two halves.

Formulas for Deriving the Area of an Isosceles Triangle

The simplest and common formula for determining the area of an isosceles type triangle is ½ x base x height. A list of other equations utilized for calculating the area is mentioned below.

Different Formulas based on various conditions:
When the dimensions of the 2 sides and the degree of an angle between them is given	A = ½ × j × k × sin(a)
When two angles and length of the side between them is provided	A = [k²×sin(b)×sin(a)/ 2×sin(2π−a−b)]
When the isosceles triangle is a right-angled triangle	A = ½ × i²
When all the three sides are given	A = ½ [√ (i² − j² ⁄4) × j]

Derivation for Isosceles Triangle Area Using Heron’s Formula

Heron’s formula aids in figuring out the formula of an isosceles triangle where only the sides of the figure are provided. There are simple steps to perform this derivation which are mentioned below:

Heron’s formula states that,

Area of a triangle with all three sides = √[s(s−k) (s−l) (s−m)]

Where, s = ½ (k + l + m) and k, l and m are the lengths of the sides of the triangle KLM

Now, for an isosceles triangle, two sides are equal therefore l = m

s = ½ (k + l + l)

⇒ s = ½ (2l + k)

Or, s = l + (k/2)

Using Heron’s formula,

Area = √[s(s−k) (s−l) (s−l)]

Or, Area = √ [s (s−l)2 (s−k)]

⇒ Area = (s−l) × √ [s (s−k)]

Replacing the value of ‘s’ we get area of the given isosceles triangle KLM

= (l + k/2 − l) × √ [(l + k/2) × ((l + k/2) − k)]

=k/2 × √ [(l + k/2) × (l − k/2)]

Or, area of isosceles triangle KLM with given three sides= k/2 × √ (l² − k²/4)

Area of Isosceles Right Triangle Formula

We are already familiar with the formula: Area of standard triangles = ½ ×base × height

Now, in isosceles right triangle base = height = s

Therefore area = ½ × s × s = s²/2

Area of Isosceles Triangle Using Trigonometry

So, to determine the area of an isosceles triangle, we must be aware of the dimensions of the two sides and the angle they make with each other. Let the sides be k and l, and the angle between them be c then,

Area = ½ × k × l × sin(c)

If only a single length is given, say ‘k’ and two corresponding angles are provided, namely ‘c’ and ‘d’ then,

Area = [k²×sin(d)×sin(c)/ 2×sin (2 x pi − c − d)]