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Find the area of the shaded region in Fig. 11.10, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA, respectively of a square ABCD (Use π = 3.14)

Solution:

Given, ABCD is a square of side 12 cm

P, Q, R and S are the midpoints of the sides AB, BC, CD and DA of square ABCD.

We have to find the area of the shaded region.

From the figure,

P is the midpoint of AB.

So, AP = PB = 12/2 = 6 cm

Q is the midpoint of BC.

So, BQ = QC = 12/2 = 6 cm

R is the midpoint of CD

So, CR = DR = 12/2 = 6 cm

S is the midpoint of DA.

So, AS = SD = 12/2 = 6 cm

We observe that the area of all four sectors made by square and circle are equal.

Corresponding angle of each sector, θ = 90°

Area of shaded region = area of square - 4(area of sector)

So, area of shaded region = area of square - area of circle

Here, radius of circle = 12/2 = 6 cm

Area of square = (side)²

= 12(12)

= 144 cm²

Area of circle = πr²

= (3.14)(6)²

= (3.14)(36)

= 113.04 cm²

Area of shaded region = 144 - 113.142

= 30.96 cm²

Therefore, area of the shaded region is 30.96 cm²

✦ Try This: From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in the given figure. Find the area of the remaining portion of the square.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12

NCERT Exemplar Class 10 Maths Exercise 11.3 Problem 11

Find the area of the shaded region in Fig. 11.10, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA, respectively of a square ABCD (Use π = 3.14)

Summary:

The area of the shaded region in Fig. 11.10, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA, respectively of a square ABCD is 30.96 cm²

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