# Three Semicircles are Inscribed Inside an Equilateral Triangle

## Semicircles are inscribed in an equilateral triangle

When three semicircles are inscribed inside an equilateral triangle, then find the area between the equilateral triangle and three semicircle

## Solution to the Geometry math problem

From the figure, Blue area = Area of the triangle – Area of the semicircles

Let the radius of the semicircle be r, AQ = x and BQ = y

Connect centres of the semicircle, then

From triangle AQR

tan 60° = RQ/AQ

⇒ √3 = 2r/x

⇒ x = 2r/√3

From triangle PQB

sin 60° = PQ/BQ

⇒ ½√3 = 2r/y

⇒ y = 4r/√3

so,

x + y = 2r/√3 + 4r/√3

⇒ 2√3 = 6r/√3

⇒ 6r = 6

then we get, the radius of the circle, r = 1 cm

Area of the equilateral triangle = √3 × (2√3)²/4

⇒ Area of the equilateral triangle = √3 × 12/4

⇒ Area of the equilateral triangle = 3√3 cm²

Area of the semicircle = ½ × π × 1²

⇒ Area of the semicircle = ½ π cm²

Now, the area of the semicircle = 3π/2 cm²

Blue area = 3√3 – 3π/2 cm²