Chapter 12: Areas Related to Circles

VVI questions — the ones the board asked most (16)

1. Q1Short answer2 marksAsked: 2017, 2022, 2023

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

   

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   Model answer · area of a quadrant of a circle

   1. 1.Given the circumference of the circle is 22 cm. Formula: Circumference = 2πr Substitution: 2πr = 22, so 2 × 22/7 × r = 22 Calculation: r = 22 × 7 / (2 × 22) = 3.5 cm So the radius r = 3.5 cm.1 mark
   2. 2.A quadrant of a circle is one-fourth of the circle, so its area is (1/4)πr². Given: r = 3.5 cm Formula: Area of quadrant = (1/4)πr² Substitution: Area = (1/4) × 22/7 × 3.5 × 3.5 Calculation: Area = (1/4) × 22/7 × 12.25 = 9.625 Answer: Area of the quadrant = 9.625 cm²1 mark

   • Show the working 2πr = 22 before stating r = 3.5 cm.
   • Write the formula (1/4)πr² before substituting values for the quadrant's area.
   • End with the final answer 9.625 cm² with its unit.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Short answer2 marksAsked: 2022, 2026

   If the radius of a circle is 21 cm and central angle of the sector is 120 degree then find the area of the sector.

   

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   Model answer · area of a sector of a circle

   1. 1.The area of a sector of a circle of angle θ is given by the formula: Area of the sector = θ/360° × πr², where r is the radius of the circle and θ the angle of the sector in degrees.1 mark
   2. 2.Given: radius r = 21 cm, angle θ = 120° Formula: Area of the sector = θ/360° × πr² Substitution: Area = 120/360 × 22/7 × 21 × 21 cm² Calculation: Area = 462 cm² Answer: The area of the sector is 462 sq cm.1 mark

   • Start by writing the formula Area of the sector = θ/360° × πr² before substituting values.
   • Show the substitution step with r = 21 cm and θ = 120° clearly before calculating.
   • End with the final answer 462 cm² along with its unit sq cm.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Long answer5 marksAsked: 2025

   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 circumferences of the two circles.

   

   Medha Ma’am

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   Model answer · circumference of circles, combined

   1. 1.Let the radius of the first circle be r1 = 19 cm. The circumference of a circle = 2πr. So, the circumference of the first circle = 2πr1 = 2π(19) = 38π cm.1 mark
   2. 2.Let the radius of the second circle be r2 = 9 cm. The circumference of the second circle = 2πr2 = 2π(9) = 18π cm.1 mark
   3. 3.Let R be the radius of the new circle whose circumference is equal to the sum of the circumferences of the two circles. So, 2πR = 2πr1 + 2πr2 = 38π + 18π = 56π cm.1.5 marks
   4. 4.From 2πR = 56π, dividing both sides by 2π, we get R = 56π / 2π.1 mark
   5. 5.On simplifying, the radius of the new circle is R = 28 cm.0.5 marks

   • Write the formula 'circumference of a circle = 2πr' before substituting values.
   • Clearly label the two given radii as r1 and r2 and the required radius as R.
   • Show the step 2πR = 2πr1 + 2πr2 before solving for R, and end with R = 28 cm.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Long answer5 marksAsked: 2026

   A circular disc of radius 6 cm is divided into three sectors with central angles 90°, 120° and 150° then what part of the whole disc is the sector with central angle 150° ? Also, calculate the areas of the three sectors.

   How the model answer begins

   1. The fraction of the whole disc represented by the sector with central angle 150° is found using the formula for area of a sector of a circle with radius r and angle with degree measure θ, which is πr²(θ/360°). Since the…

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5. Q5Short answer2 marksAsked: 2025

   The length of the minute hand for a clock is 7 cm. Find the area swept by it in 40 minutes.

   How the model answer begins

   1. The angle subtended by the minute hand in 60 minutes is 360°, so the angle subtended in 40 minutes is θ = (360/60) × 40 = 240°.

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6. Q6Short answer2 marksAsked: 2026

   Find the circumference and area of a circle having diameter 28 cm.

   How the model answer begins

   1. Radius r = 28/2 = 14 cm; circumference = 2πr = 2 × 22/7 × 14 = 88 cm.

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7. Q7Short answer2 marksAsked: 2022

   A wire is in the form of a circle of radius 84 cm. Find the side of a square which can be made by this wire.

   How the model answer begins

   1. The circumference of the circle is given by the formula 2πr, where r is the radius of the circle. Here, radius r = 84 cm. So, circumference = 2 × 22/7 × 84 = 528 cm. Since the wire is in the form of this circle, the…

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8. Q8Short answer2 marksAsked: 2022

   Find the perimeter of a semicircular field whose diameter is 24 cm.

   How the model answer begins

   1. The perimeter of a semicircular field is given by the formula πr + 2r, where r is the radius of the circle. This is because the perimeter consists of the curved arc of length πr and the diameter (straight edge) of…

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9. Q9Short answer2 marksAsked: 2022

   Find the area of a ring whose outer and inner radii are 23 cm and 21 cm respectively.

   How the model answer begins

   1. The area of a ring = π(R² − r²), where R = 23 cm is the outer radius and r = 21 cm the inner radius.

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10. Q10Short answer2 marksAsked: 2023

    If the perimeter of a semi-circular protractor is 36 cm, then find its diameter.

    How the model answer begins

    1. The perimeter of a semi-circular protractor of radius r is πr + 2r, which is the sum of the length of the arc and the two straight radii forming its diameter. Here the perimeter is given as 36 cm, so πr + 2r = 36, which…

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11. Q11Short answer2 marksAsked: 2023

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

    How the model answer begins

    1. The circumference of the circle is given as 22 cm. Circumference of a circle = 2πr So, 2πr = 22 2 × 22/7 × r = 22 r = 22 × 7 / (2 × 22) r = 7/2 cm = 3.5 cm

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12. Q12Short answer2 marksAsked: 2023

    If the perimeter of a semi-circular protractor is 36 cm, then find its diameter.

    How the model answer begins

    1. The perimeter of a semi-circular protractor is given by πr + 2r, where r is the radius of the protractor. Here the perimeter is 36 cm, so πr + 2r = 36, which can be written as r(π + 2) = 36.

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13. Q13Short answer2 marksAsked: 2021

    What will be the length of the arc of a circle with radius 21 cm that subtends an angle of 60 degree at the center of the circle?

    How the model answer begins

    1. The length of an arc of a sector of angle θ is given by the formula (θ/360) × 2πr, where r is the radius of the circle. Here, radius r = 21 cm and the angle subtended at the centre θ = 60°. Substituting these values…

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14. Q14Short answer2 marksAsked: 2021

    If the side of an equilateral triangle is 2a units, find its area.

    How the model answer begins

    1. The area of an equilateral triangle is given by the formula Area = √3/4 × (side)².

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15. Q15Short answer2 marksAsked: 2021

    The length of the hour hand of a clock is 6 cm. Find the area swept by this hand in 4 hours.

    How the model answer begins

    1. The hour hand of a clock makes an angle of 360 degrees in 12 hours. So the angle made by the hour hand in 4 hours is (360/12) × 4 = 120 degrees. Here the length of the hour hand is r = 6 cm, which acts as the radius of…

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16. Q16Short answer2 marksAsked: 2021

    An arc of a circle of radius 21 cm subtends an angle of 120° at the centre, find the length of the arc.

    How the model answer begins

    1. The length of an arc of a sector of angle θ = θ/360 × 2πr, where r is the radius of the circle and θ the angle subtended at the centre in degrees. Here, radius r = 21 cm and θ = 120°, so the length of the arc l =…

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Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   If the ratio of circumferences of two circles is 3 : 4 then the ratio of their areas is

   • (A) 9 : 16 ✓
   • (B) 16 : 9
   • (C) 4 : 3
   • (D) 3 : 4

   Answer: (A)

2. 2. Bihar Board 2026

   The value of central angle of a sector is 60° and radius is 21 cm. Then the length of the arc is

   • (A) 11 cm
   • (B) 21 cm
   • (C) 22 cm ✓
   • (D) 44 cm

   Answer: (C)

3. 3. Bihar Board 2026

   Perimeter of semicircle is

   • (A) pi r
   • (B) 2pi r
   • (C) pi r + 2r ✓
   • (D) pi r + r

   Answer: (C)

4. 4. Bihar Board 2026

   Circumference of the circle having diameter 28 cm is

   • (A) 22 cm
   • (B) 44 cm
   • (C) 88 cm ✓
   • (D) 176 cm

   Answer: (C)

5. 5. Bihar Board 2026

   The perimeter of a sector of a circle of radius 14 cm is 50 cm ; then central angle of the sector is

   • (A) 36°
   • (B) 45°
   • (C) 72°
   • (D) 90° ✓

   Answer: (D)

6. 6. Bihar Board 2026

   A wire is looped in the form of a circle of radius 28 cm. It is rebent into a square form, then the length of its side is

   • (A) 11 cm
   • (B) 22 cm
   • (C) 44 cm ✓
   • (D) 88 cm

   Answer: (C)

7. 7. Bihar Board 2026

   The area of a circle is the same as the area of a square. Then the ratio of their perimeters is

   • (A) 2 : pi
   • (B) pi : 2
   • (C) sqrt(pi) : 2 ✓
   • (D) 2 : sqrt(pi)

   Answer: (C)

8. 8. Bihar Board 2026

   If the circumference of a circle increases from 2pi to 4pi then its new area is

   • (A) halved
   • (B) doubled
   • (C) three times
   • (D) four times ✓

   Answer: (D)

9. 9. Bihar Board 2025

   The ratio of the radii of two circles is 3 : 4 ; then the ratio of their areas is

   • (A) 3 : 4
   • (B) 4 : 3
   • (C) 9 : 16 ✓
   • (D) 16 : 9

   Answer: (C)

10. 10. Bihar Board 2025

    The area of the sector of a circle of radius 42 cm and central angle 30° is

    • (A) 515 cm^2
    • (B) 416 cm^2
    • (C) 462 cm^2 ✓
    • (D) 406 cm^2

    Answer: (C)

Questions parents and students ask

How many marks does "Areas Related to Circles" carry in Bihar Board Class 10 Maths?

In the Bihar Board papers of 2021–2026, this chapter carried about 5 marks and about 6 questions a year; by marks it ranks 11 in Maths.

Which are the most important (VVI) questions of "Areas Related to Circles"?

The 16 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 16 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.