Three Semicircles are Inscribed Inside an Equilateral Triangle
![When three semicircles are inscribed inside an equilateral triangle, then find the area between the equilateral triangle and three semicircle](https://aplusreach.com/wp-content/uploads/2021/12/Geometry-Math-Problem-Three-Semicircles-are-Inscribed-Inside-an-Equilateral-Triangle-1024x683.jpg)
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
![](https://aplusreach.com/wp-content/uploads/2021/12/Geometry-Math-Problem-Three-Semicircles-are-Inscribed-Inside-an-Equilateral-Triangle-1-1024x683.jpg)
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
![](https://aplusreach.com/wp-content/uploads/2021/12/Geometry-Math-Problem-Three-Semicircles-are-Inscribed-Inside-an-Equilateral-Triangle-2-1-1024x683.jpg)
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