Question

Two semicircles and one circle of same radius 5 cm are inscribed in a rectangle. Find the area of shaded region. [Take π = 3.14]

Answer 1

We are given:

• A rectangle that contains:
  • Two semicircles on each end
  • One full circle in the center
• All three have the same radius = 5 cm
• We are to find the shaded area i.e. area of rectangle minus area occupied by the curves
• Use π = 3.14

Step-wise calculation:

Step 1: Determine the dimensions of the rectangle

The rectangle’s height is equal to the diameter of the circle or semicircles, which is twice the radius.

Height of rectangle = 2r

= 2 × 5 cm

= 10 cm

The rectangle’s length is equal to the diameter of the two semicircles plus the diameter of the full circle. 

Since two semicircles effectively form one full circle, the length is the sum of three diameters.

Length of rectangle = r + 2r + r

= 4r

= 4 × 5 cm

= 20 cm

Step 2: Calculate the area of the rectangle

Area of rectangle = Length × Height

= 20 cm × 10 cm

= 200 cm²

Step 3: Calculate the area of the unshaded regions

The unshaded regions consist of two semicircles and one full circle.

The combined area of the two semicircles is equal to the area of one full circle. 

Therefore, the total unshaded area is equivalent to the area of two full circles.

Area of one circle = πr²

= 3.14 × 5 cm²

= 3.14 × 25 cm²

= 78.5 cm²

Total area of unshaded regions = 2 × Area of one circle

= 2 × 78.5 cm²

= 157 cm2²

Step 4: Calculate the area of the shaded region

The area of the shaded region is the area of the rectangle minus the total area of the unshaded regions.

Area of shaded region = Area of rectangle - Total area of unshaded regions

Area of shaded region = 200 cm² – 157 cm² = 43 cm².

he area of the shaded region is 43 cm².