In the given figure ΔABC is right angled at A. Semicircles are drawn on AB, AC and BC as diameter. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region.

Let semicircle I, II and III are semicircles with diameters AB, AC and BC respectively

Area of shaded region =

Area of semicircle I + Area of semicircle II + Area of triangle ABC – Area of semicircle III

As, ∠BAC is in semicircle,

∠BAC = 90° [Angle in a semicircle is right angle]

And ABC is a right – angled triangle at A

By Pythagoras Theorem

(Hypotenuse)² = (Base)² + (Perpendicular)²

(BC)² = (AB)² + (AC)²

⇒ (BC)² = 3² + 4² = 9 + 16 = 25

⇒ BC = 5 cm

Now, For semicircle I

Diameter = AB = 3 cm

Radius

Area of semicircle of radius r

Area of semicircle I

For semicircle II

Diameter = AC = 4 cm

Radius

Area of semicircle of radius r

Area of semicircle II

For semicircle III

Diameter = BC = 5 cm

Radius,

Area of semicircle of radius r

Area of semicircle I

Area of a right – angled triangle

Area of ΔABC

Required area (From eqn [1])