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Areas Related to Circles

Short Answer Type

51. The area of an equilateral triangle is

49 square root of 3 space cm squared.

Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of triangle not included in the circles.

open square brackets bold Take bold space square root of bold 3 bold equals bold 1 bold. bold 73 close square brackets

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52.
Fig. 12.47A shows a decorative block which is made of two solids—a cube and a hemisphere. The base of the block is a cube with edge 5 cm and the hemisphere, fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block.

$open square brackets Take space straight pi space equals 22 over 7 close square brackets$

The total surface area of the cube
= 6 × (edge)²= 6 × 5 × 5 cm²= 150 cm²Note that the part of the cube where the hemisphere is attached in not included in the surface area.
∴ The surface area of the block
= Total surface area of cube - base area of hemisphere + curved surface area of hemisphere

$equals space 150 minus πr squared plus 2 πr squared equals space 150 plus πr squared equals space open parentheses 150 plus 22 over 7 straight x fraction numerator 4.2 over denominator 2 end fraction straight x fraction numerator 4.2 over denominator 2 end fraction close parentheses space cm squared equals space open parentheses 150 plus 22 over 7 straight x 42 over 20 42 over 20 close parentheses space cm squared equals space open parentheses 150 plus fraction numerator 33 space straight x space 42 over denominator 100 end fraction close parentheses space cm squared equals space open parentheses 150 plus 1386 over 100 close parentheses space cm squared equals space left parenthesis 150 space plus space 13.86 right parenthesis space cm squared equals space 163.86 space cm squared$

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53. Find the area of a sector of angle 0 of a circle with radius R.

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54. If the perimeter and the area of a circle are numerically equal. Find the radius of the circle.

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55. Find the area of a ring with outer and inner radii being R and r.

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56. A steel wire, when bent in the form of a square, enclosed an area of 121 cm^;. The same wire is bent in the form of a circle. Find the area of the circle.

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