# In Fig. 12.25, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region

**Solution:**

We use the concepts of areas of circles and squares to solve the problem.

Since the circles are touching each other externally, visually it is clear that the radius of each circle, r = 1/2 × (side of the square)

Also, ABCD being a square all angles are of measure 90°. Therefore, all sectors are equal as they have the same radii and angle.

∴ The angle of each sector which is part of the square (θ) = 90°

∴ Area of each sector = θ/360° × πr^{2}

= 90°/360° × πr^{2}

= πr^{2}/4

From the figure, it is clear that

Area of shaded region = Area of square - Area of 4 sectors

= (side)^{2} - 4 × Area of each sector

= (14)^{2} - 4 × πr^{2}/4

Area of each of the 4 sectors is equal as each sector subtends an angle of 90° at the center of a circle with radius, r = 1/2 × 14 cm = 7 cm

Area of each sector = θ/360° × πr^{2}

= 90°/360° × π (7)^{2}

= 1/4 × 22/7 × 7 cm × 7 cm

= 77/2 cm^{2}

Area of shaded region = Area of square - 4 × Area of each sector

= (14)² cm^{2 }- 4 × 77/2 cm^{2}

= 196 cm^{2 }- 154 cm^{2}

= 42 cm^{2}

**Video Solution:**

## In Fig. 12.25, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region

### NCERT Solutions Class 10 Maths - Chapter 12 Exercise 12.3 Question 7:

In Fig. 12.25, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region

The area of the shaded region of a square ABCD of side 14 cm where four circles are drawn with A, B, C, D as centers such that each circle touch externally two of the remaining three circles is 42 cm^{2}