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Areas Related to Circles

Short Answer Type

1. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumference of two circles.

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2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.

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Long Answer Type

3. Fig. 12.3, depicts an archery target marked with its five scoring areas from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

Fig. 12.3

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Short Answer Type

4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

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Tick the correct answer in the following: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is
(A) 2 units (D) π units
(C) 4 units (D) 7 units.

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6. Find the area of a sector of a circle with radius 6 cm and of angle 60°.

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7. Find the area of a quadrant of a circle whose circumference is 22 cm.

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8. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

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A chord of circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding :

(i) Minor segment
(ii) Major sector.

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10.
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find :

(i)    the length of the arc
(ii)    area of the sector formed by the arc
(iii)    area of the segment formed by the corresponding chord.

Here, we have
r = 21 cm and ө = 60°
Let OACBO is the given sector and $ACB with overbrace on top$ is the length of arc, then,

Here, we haver = 21 cm and ө = 60°Let OACBO is the given sector and
(i) The length of the arc $open parentheses ACB with overbrace on top close parentheses$

$equals 2 πr cross times straight theta over 360 equals space open parentheses 2 cross times 22 over 7 cross times 21 cross times fraction numerator 60 over denominator 360 degree end fraction close parentheses space cm squared equals space 22 space cm squared$

(ii) Area of the sector formed by the arc (OACBO)

$equals 2 πr cross times straight theta over 360 equals space open parentheses 2 cross times 21 cross times 21 cross times fraction numerator 60 over denominator 360 degree end fraction close parentheses space cm squared equals space 231 space cm squared$

(iii) Area of Minor segment (ACBA)

= Area of sector (OACBO) - Area of triangle (OAB)

$equals space πr squared cross times space straight theta over 360 minus 1 half straight r squared space sin space straight theta equals space 231 minus 1 half straight x space space 21 space straight x space 21 space straight x space sin space 60 degree equals open parentheses 231 minus 1 half straight x space space 21 space straight x space 21 space straight x space fraction numerator square root of 3 over denominator 2 end fraction close parentheses space cm squared equals space open parentheses 231 minus fraction numerator 441 square root of 3 over denominator 4 end fraction close parentheses space cm squared.$