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 (θ) = 90° [Since it is a part of a square]

∴ Area of each sector = θ/360° × πr²

= 90°/360° × πr²

= πr²/4

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

= 1/4 × 22/7 × 7 cm × 7 cm

= 77/2 cm²

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

= (14)² cm² - 4 × 77/2 cm²

= 196 cm² - 154 cm²

= 42 cm²

☛ Check: NCERT Solutions for Class 10 Maths Chapter 12

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

Summary:

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 cm².

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