Find the shaded area of Geometric Shape Formed by Semicircles and a Circle

A circle and two semicircles are inscribed inside a semicircle

Find the shaded area inside a semicircle, when a circle and two semicircles are inscribed inside a semicircle and connect a semicircle to the bigger semicircle, (The diameter of the circle is 6 cm)

Solution

From the figure

Blue area = Area of the bigger semicircle – Area of the circle – Area of the 1st and 2nd smaller semicircles + Area of the 3rd smaller semicircle

The radius of the 1st semicircle and 2nd semicircle is equal (because of the symmetry)

Now connect the centre of the semicircle and the centre of the 1st semicircle

Let the radius of the 1st semicircle = r, then

AR = RN = r

PN = AP – (AR + RN) = 6 – 2r

Apply Pythagoras theorem in triangle PQR

QR² = PR² + PQ²

⇒ (3 + r)² = (6 – r)² + 3²

⇒ 9 + 6r + r² = 36 – 12r + r² + 9

so, 6r = – 12r + 36

⇒ 18r = 36

⇒ r = 2 cm

Now Radius of the 3rd semicircle = 6 – 2r = 2 cm

That is the radius of three smaller semicircles are equal

then, the area of the smaller semicircle = ½ × π × 2²

⇒ Area of the smaller semicircle = 2π cm²

Area of the circle = π × 3²

⇒ Area of the circle = 9π cm²

Area of the bigger semicircle = ½ × π × 6²

⇒ Area of the bigger semicircle = 18π cm²

Blue area = 18π – 9π – (2π + 2π) + 2π

⇒ Blue area = 7π cm²