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