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Areas Related to Circles

Short Answer Type

21. Find the area of the shaded region in fig. 12.20 if radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and ∠ AOC = 40°.

Fig. 12.20

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22.

Find the area of the shaded region in Fig. 12.21, if ABCD is a square of side 14 cm and APD and BPC are semicircles.

Fig. 12.21.

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23. Find the area of the shaded region in Fig. 12.22, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

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24.
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 Fig. 12.23. Find the area of the remaining portion of the square.

Fig. 12.23

We have,
(i) Side of square = 4 cm
∴ Area = (side)² = 16 cm²(ii) Diameter of circle = 2 cm

We have,(i) Side of square = 4 cm∴ Area = (side)2 = 16 cm2(ii) Dia

So, radius = 1 cm
And, area = $πr squared$
$equals space open parentheses 22 over 7 straight x space 1 space straight x 1 close parentheses space cm squared equals space 22 over 7 space cm squared$
(iii) Area of 4 quadrants of radius 1 cm

$equals space 4 space straight x space 1 fourth space πr squared space equals space πr squared equals space open parentheses 22 over 7 straight x space 1 space straight x space 1 close parentheses space cm squared equals space 22 over 7 space cm squared$
Now required area :
= Area of square - (Area of 4 quadrants + Area of circle)
$equals space open square brackets 16 minus open parentheses 22 over 7 plus 22 over 7 close parentheses close square brackets space cm squared space equals space open square brackets 16 minus 44 over 7 close square brackets space cm squared equals space open square brackets fraction numerator 112 minus 44 over denominator 7 end fraction close square brackets space cm squared equals space 68 over 7 space cm squared.$

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Long Answer Type

25.

In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Figure. Find the area of the design

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Short Answer Type

26. 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.

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27.

Fig. 12.26 depicts a racing track whose left and right ends are semicircular.

Fig. 12.26

The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find :
(i) the distance around the track along its inner edge.
(ii) the area of the track.

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28.

In Fig. 12.27, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

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Long Answer Type

29.

The area of an equilateral triangle ABC is 17320.5 cm2 . With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Fig. 12.28). Find the area of the shaded region (Use $straight pi$ = 3.14 and $square root of 3 equals$ 1.73205)

Fig. 12.28

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Short Answer Type

30.

On a square handkerchief, nine circular designs each of radius 7 cm are made (see Fig. 12.29). Find the area of the remaining portion of the handkerchief.

Fig. 12.29