Area Related to Circles

A man wants to fence a plot in the shape of a sector of a circle of radius r metres with a central angle of π/3 radians. If the total length of the fencing available is 50 metres, what should be the maximum radius for the sector?

1. 120(π - 6)
2. 150/(π + 6)
3. 50/(π + 6)
4. 110(π - 3)

Answer: (b) 150/(π + 6)

If the area of a circle with a certain radius increases by 31.4 cm2 upon increasing the radius by 1 cm, what would be the increase in circumference?

1. 6.54 cm
2. 1.15 cm
3. 6.28 cm
4. 12.56 cm

Answer: (c) 6.28 cm

In the following figure, PT = 4 cm, TQ = 6 cm and RT = 3 cm, then what is the value of TS?

1. 8 cm
2. 5 cm
3. 2 cm
4. 4 cm

Answer: (a) 8 cm

Four identical coins are placed in a square. For each coin, the ratio of area to circumference is the same ratio of circumference to area. Then, find the area of the square that is not covered by the coins.

1. 16(π - 1) cm2
2. 16(8 - π) cm2
3. 16(4 - π) cm2
4. 16(4 - π/2) cm2

Answer: (c) 16(4 - π) cm2

ABC is a right-angled triangle, right angled at B. BC = 21 cm and AB = 28 cm. Width AC as the diameter of a semicircle and width BC as the radius of a quarter circle D are drawn. What is the area of the shaded portion?

1. 425.5 cm2
2. 420 cm2
3. 428.75 cm2
4. 114 cm2

Answer: (c) 428.75 cm2

If the sides 50 m and 130 m of a triangular field meet at an angle of 72o, then find the area in which wheat is cultivated (shaded portion).

1. 115π m2
2. 120π m2
3. 155π m2
4. 180π m2

Answer: (d) 180π m2

Let C be a circle with centre P0 and AB be the diameter of C. Suppose P1 is the midpoint of the line segment P0B, P2 is the midpoint of the line segment P1B and so on. Let C1, C2, C3,…... be the circles with diameters P0P1, P1P2, P2P3,….. respectively. Suppose the circles C1, C2, C3... are all shaded. What is the ratio of the area of the unshaded portion of C to the original circle C?

1. 11 : 12
2. 12 : 05
3. 4 : 3
4. 5 : 6

Answer: (a) 11 : 12

Three circles of radius 2 cm touch one another externally. These circles are circumscribed by a circle of radius R cm. Find the area of the shaded region.

1. (4√3 + 1) cm2
2. 4π/3((4√3 + 1) - 4√3) cm2
3. 4π/3( - 4√3) cm2
4. 4π/3((4√3 + 1) cm2

Answer: (b) 4π/3((4√3 + 1) - 4√3) cm2

Read the passage and answer the following question: The diagram consists of a sector of a circle, a triangle, and a rectangle. The sector has centre O, a radius of x cm, and an angle of 270∘. The rectangle has a length of 2x cm and is attached to the base of the triangle. The triangle is formed by the two radii of the sector and a chord at the bottom. The total area of the shape is expressed as kx2 cm². A new sector of radius 2x is added to the shape, attached to the rectangle's bottom edge with an angle of 180o. What is the total area of the entire shape?

1. kx2 - 2πx
2. kx - 2πx2
3. 2πx2
4. kx2 + 2πx2

Answer: (d) kx2 + 2πx2

The diagram shows a circle with radius 1 that rolls without slipping around the inside of a square with sides of length 5. The circle rolls once around the square, returning to its starting point. What distance does the centre of the circle travel?

1. (16 - 2π)
2. 12
3. (12 + 2π)
4. 16

Answer: (b) 12