Question 1

The circumference of a circle is 88 cm. If the radius is increased by 50%, what is the percentage increase in the area of the circle?

Accepted Answer

Step 1: Original circumference = 88 cm. Using C = 2πr: 88 = 2 × (22/7) × r, so r = 14 cm. Step 2: Original area = πr² = (22/7) × 14² = 616 cm². Step 3: New radius = 14 + (50% of 14) = 14 + 7 = 21 cm. Step 4: New area = (22/7) × 21² = 1386 cm². Step 5: Percentage increase = [(1386 - 616) / 616] × 100 = (770/616) × 100 = 125%.

Question 2

A circular track has an inner radius of 35 m and outer radius of 42 m. If a runner completes 5 laps on the inner circle and another completes 4 laps on the outer circle, what is the difference in distances covered (in metres)?

Accepted Answer

Step 1: Inner circle circumference = 2πr = 2 × (22/7) × 35 = 220 m. Step 2: Distance covered by first runner = 5 × 220 = 1100 m. Step 3: Outer circle circumference = 2 × (22/7) × 42 = 264 m. Step 4: Distance covered by second runner = 4 × 264 = 1056 m. Step 5: Difference = 1100 - 1056 = 44 m.

Question 3

A sector of a circle with radius 21 cm has an arc length of 33 cm. What is the area of this sector (in cm²)?

Accepted Answer

Step 1: Arc length formula: L = (θ/360) × 2πr, where θ is in degrees. Step 2: 33 = (θ/360) × 2 × (22/7) × 21. Step 3: 33 = (θ/360) × 132, so θ = (33 × 360) / 132 = 90°. Step 4: Area of sector = (θ/360) × πr² = (90/360) × (22/7) × 21² = (1/4) × (22/7) × 441 = (1/4) × 1386 = 346.5 cm².

Question 4

Two circles have areas in the ratio 4:9. If the circumference of the smaller circle is 44 cm, what is the circumference of the larger circle (in cm)?

Accepted Answer

Step 1: Area ratio = 4:9, so radius ratio = √4 : √9 = 2:3. Step 2: Let radii be 2k and 3k. Circumference of smaller circle = 2π(2k) = 44 cm. Step 3: 4πk = 44, so πk = 11, thus k = 11/π. Step 4: Radius of larger circle = 3k = 33/π. Step 5: Circumference of larger circle = 2π(3k) = 6πk = 6 × 11 = 66 cm.

Question 5

A circular garden has a radius of 28 m. A path of uniform width 3.5 m is constructed around it on the outside. If the cost of constructing the path is ₹150 per m², what is the total cost (in ₹)?

Accepted Answer

Step 1: Inner radius (garden) = 28 m. Step 2: Outer radius (including path) = 28 + 3.5 = 31.5 m. Step 3: Area of path = π(R² - r²) = (22/7) × (31.5² - 28²). Step 4: 31.5² = 992.25, 28² = 784. Step 5: Area of path = (22/7) × (992.25 - 784) = (22/7) × 208.25 = (22/7) × (833/4) = (22 × 833) / 28 = 18326/28 = 654.5 m². Step 6: Total cost = 654.5 × 150 = ₹98,175.

Question 6

A circular garden has a radius of 21 m. What is the cost of fencing it at ₹5 per metre? (Use π = 22/7)

Accepted Answer

Step 1: Calculate the circumference using C = 2πr. Step 2: C = 2 × (22/7) × 21 = 2 × 22 × 3 = 132 m. Step 3: Cost = Circumference × Rate per metre = 132 × 5 = ₹660. Therefore, the total cost is ₹660.

Question 7

A circular garden has an area of 616 m². What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Apply the area formula A = πr². Step 2: Substitute A = 616 and π = 22/7. Step 3: 616 = (22/7) × r². Step 4: r² = 616 × (7/22) = 196. Step 5: r = √196 = 14 m. Therefore, the radius is 14 m.

Question 8

The area of a circle is 154 cm². What is its circumference? (Use π = 22/7)

Accepted Answer

Step 1: From A = πr², find r: 154 = (22/7) × r². Step 2: r² = 154 × 7/22 = 49, so r = 7 cm. Step 3: Use C = 2πr = 2 × (22/7) × 7 = 44 cm.

Question 9

Two circles have radii 5 cm and 12 cm respectively. What is the difference in their areas? (Use π = 3.14)

Accepted Answer

Step 1: Calculate area of first circle: A₁ = πr₁² = 3.14 × 5² = 3.14 × 25 = 78.5 cm². Step 2: Calculate area of second circle: A₂ = πr₂² = 3.14 × 12² = 3.14 × 144 = 452.16 cm². Step 3: Find the difference: A₂ - A₁ = 452.16 - 78.5 = 373.66 cm². Therefore, the difference is 373.66 cm².

Question 10

The circumference of a circle is 88 cm. What is its area? (Use π = 22/7)

Accepted Answer

Step 1: Use circumference formula C = 2πr to find radius. Step 2: 88 = 2 × (22/7) × r. Step 3: r = 88 × 7 / (2 × 22) = 14 cm. Step 4: Apply area formula A = πr². Step 5: A = (22/7) × 14² = (22/7) × 196 = 616 cm². Therefore, the area is 616 cm².

Question 11

A wheel has a radius of 35 cm. How many complete revolutions will it make to cover a distance of 2200 m? (Use π = 22/7)

Accepted Answer

Step 1: Calculate the circumference of the wheel using C = 2πr. Step 2: C = 2 × (22/7) × 35 = 220 cm. Step 3: Convert distance to same units: 2200 m = 220,000 cm. Step 4: Number of revolutions = Total distance / Circumference = 220,000 / 220 = 1000. Therefore, the wheel makes 1000 complete revolutions.

Question 12

The circumference of a circle is 88 cm. What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Use the formula C = 2πr. Step 2: Substitute 88 = 2 × (22/7) × r. Step 3: Solve for r: 88 = (44/7) × r, so r = 88 × 7/44 = 14 cm.

Question 13

A circular track has a circumference of 440 m. A runner completes 2.5 laps. What distance does the runner cover?

Accepted Answer

Step 1: Distance per lap = circumference = 440 m. Step 2: Distance for 2.5 laps = 2.5 × 440 = 1100 m.

Question 14

A circle has a diameter of 14 m. Find its area. (Use π = 22/7)

Accepted Answer

Step 1: Find the radius from diameter: r = d/2 = 14/2 = 7 m. Step 2: Use the area formula: A = πr². Step 3: A = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 m². Therefore, the area is 154 m².

Question 15

The area of a circle is 616 cm². What is its circumference? (Use π = 22/7)

Accepted Answer

Step 1: Use area formula A = πr² to find radius: 616 = (22/7) × r². Step 2: r² = 616 × (7/22) = 28 × 7 = 196. Step 3: r = √196 = 14 cm. Step 4: Calculate circumference: C = 2πr = 2 × (22/7) × 14 = 2 × 22 × 2 = 88 cm. Therefore, the circumference is 88 cm.

Question 16

The radius of a circle is 7 cm. What is its circumference? (Use π = 22/7)

Accepted Answer

Step 1: Use the formula for circumference: C = 2πr. Step 2: Substitute r = 7 cm and π = 22/7. Step 3: C = 2 × (22/7) × 7 = 2 × 22 = 44 cm. Therefore, the circumference is 44 cm.

Question 17

The circumference of a circle is 44 cm. What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Use the circumference formula: C = 2πr. Step 2: Rearrange to find r: r = C/(2π). Step 3: r = 44/(2 × 22/7) = 44/(44/7) = 44 × (7/44) = 7 cm. Therefore, the radius is 7 cm.

Question 18

Two circles have radii 7 cm and 14 cm respectively. What is the ratio of their areas?

Accepted Answer

Step 1: Area of circle 1: A₁ = π × 7² = 49π. Step 2: Area of circle 2: A₂ = π × 14² = 196π. Step 3: Ratio = 49π : 196π = 49 : 196 = 1 : 4.

Question 19

A wheel has a diameter of 35 cm. How many complete revolutions will it make to cover a distance of 1100 m?

Accepted Answer

Step 1: Find circumference using C = πd = (22/7) × 35 = 110 cm. Step 2: Convert distance to cm: 1100 m = 110,000 cm. Step 3: Number of revolutions = 110,000 ÷ 110 = 1000.

Question 20

A circular garden has a radius of 21 m. What is its area? (Use π = 22/7)

Accepted Answer

Step 1: Use the formula A = πr². Step 2: Substitute A = (22/7) × 21². Step 3: Calculate 21² = 441. Step 4: A = (22/7) × 441 = 22 × 63 = 1386 m².

Question 21

A wheel has a diameter of 70 cm. How many complete revolutions will it make to cover a distance of 2.2 km?

Accepted Answer

Step 1: Diameter = 70 cm, so radius = 35 cm. Step 2: Circumference = πd = (22/7) × 70 = 220 cm = 2.2 m. Step 3: Distance to cover = 2.2 km = 2200 m. Step 4: Number of revolutions = Distance / Circumference = 2200 / 2.2 = 1000 revolutions.

Question 22

A circular garden has a circumference of 88 metres. What is the area of the garden in square metres? (Use π = 22/7)

Accepted Answer

Step 1: Use circumference formula C = 2πr to find radius. 88 = 2 × (22/7) × r. Step 2: Solve for r: r = 88 × 7 / (2 × 22) = 616 / 44 = 14 metres. Step 3: Calculate area using A = πr². A = (22/7) × 14² = (22/7) × 196 = 22 × 28 = 616 square metres.

Question 23

A wheel has a radius of 35 cm. How many complete revolutions will it make to cover a distance of 2.2 km? (Use π = 22/7)

Accepted Answer

Step 1: Calculate the circumference of the wheel using C = 2πr. C = 2 × (22/7) × 35 = 2 × 22 × 5 = 220 cm. Step 2: Convert distance to cm: 2.2 km = 2.2 × 100,000 cm = 220,000 cm. Step 3: Number of revolutions = Distance / Circumference = 220,000 / 220 = 1000 revolutions.

Question 24

A circle has an area of 154 square metres. If the radius is increased by 50%, what will be the percentage increase in the area? (Use π = 22/7)

Accepted Answer

Step 1: From A = πr², find original radius: 154 = (22/7) × r², so r² = 154 × 7/22 = 49, thus r = 7 m. Step 2: New radius = 7 + (50% of 7) = 7 + 3.5 = 10.5 m. Step 3: New area = (22/7) × (10.5)² = (22/7) × 110.25 = 346.5 sq. m. Step 4: Percentage increase = [(346.5 − 154) / 154] × 100 = (192.5 / 154) × 100 = 125%.

Question 25

Two concentric circles have radii 8 cm and 6 cm respectively. What is the area of the ring (annulus) formed between them? (Use π = 22/7)

Accepted Answer

Step 1: Use the annulus area formula A = π(R² - r²), where R = 8 cm (outer radius) and r = 6 cm (inner radius). Step 2: A = (22/7)(8² - 6²) = (22/7)(64 - 36) = (22/7) × 28. Step 3: A = 22 × 4 = 88 cm².

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