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Sectors and arcs practice questions with answers

Foundation and Higher · Grade 5

A second set of sectors and arcs questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

Work out the area of a sector with angle 120 degrees and radius 3. Give your answer in terms of π.

Show answer

Answer: 3π

  1. Fraction: 120/360 = 1/3.
  2. Area: (1/3) × 9π = 3π.
  • Method (1): Fraction: 120/360 = 1/3.
  • Accuracy (1): Area: (1/3) × 9π = 3π.

Question 2 Easy

Work out the area of a sector with angle 180 degrees and radius 2. Give your answer in terms of π.

Show answer

Answer: 2π

  1. Fraction: 180/360 = 1/2.
  2. Area: (1/2) × 4π = 2π.
  • Method (1): Fraction: 180/360 = 1/2.
  • Accuracy (1): Area: (1/2) × 4π = 2π.

Question 3 Medium

Work out the arc length of a sector with angle 180 degrees and radius 3. Give your answer in terms of π.

Show answer

Answer: 3π

  1. Fraction: 180/360 = 1/2.
  2. Circumference: (1/2) × 6π = 3π.
  • Method (1): Fraction: 180/360 = 1/2.
  • Accuracy (1): Circumference: (1/2) × 6π = 3π.

Question 4 Medium

What fraction of a circle is a 60 degree sector?

Show answer

Answer: 1/6

  1. Angle over 360: 60 / 360.
  2. Simplify: 1/6.
  • Method (1): Angle over 360: 60 / 360.
  • Accuracy (1): Simplify: 1/6.

Question 5 Hard

Work out the perimeter of a sector with angle 60 degrees and radius 6. Give your answer in terms of π.

Show answer

Answer: 12 + 2π

  1. Arc: (60/360) × 12π = 2π.
  2. Two radii: 6 + 6 = 12, total 12 + 2π.
  • Method (1): Arc: (60/360) × 12π = 2π.
  • Accuracy (1): Two radii: 6 + 6 = 12, total 12 + 2π.

Question 6 Hard

Work out the perimeter of a sector with angle 45 degrees and radius 4. Give your answer in terms of π.

Show answer

Answer: 8 + π

  1. Arc: (45/360) × 8π = π.
  2. Two radii: 4 + 4 = 8, total 8 + π.
  • Method (1): Arc: (45/360) × 8π = π.
  • Accuracy (1): Two radii: 4 + 4 = 8, total 8 + π.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Work out the area of a sector with angle 90 degrees and radius 4. Give your answer in terms of π.

Show answer

Answer: 4π

  1. Fraction of circle is 90 / 360 = 1/4.
  2. Full circle area is π × 4^2 = 16π, so sector area is 16π / 4 = 4π.
  • Method (1): (90/360) × π × 4^2, or 16π seen
  • Accuracy (1): 4π

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What fraction of a full circle is a 60 degree sector?

  1. 1/6
  2. 1/3
  3. 1/4
Show answer

1/6 60 / 360 = 1/6.

Check 2

What fraction of a full circle is a 90 degree sector?

  1. 1/4
  2. 1/2
  3. 1/8
Show answer

1/4 90 / 360 = 1/4.

Check 3

What fraction of a full circle is a 120 degree sector?

  1. 1/3
  2. 1/6
  3. 1/4
Show answer

1/3 120 / 360 = 1/3.

Check 4

How do you find the area of a sector of angle θ and radius r?

  1. (θ / 360) × πr^2
  2. (θ / 360) × 2πr
  3. θ × πr^2
Show answer

(θ / 360) × πr^2 Multiply the full circle area by the angle fraction.

Check 5

How do you find the arc length of a sector of angle θ and radius r?

  1. (θ / 360) × 2πr
  2. (θ / 360) × πr^2
  3. 2πr + 2r
Show answer

(θ / 360) × 2πr Multiply the circumference by the angle fraction.

Check 6

What is the area of a sector with angle 60° and radius 6 in terms of π?

  1. 6π
  2. 36π
  3. 12π
Show answer

6π (1/6) × 36π = 6π.

Check 7

What is the area of a sector with angle 90° and radius 2 in terms of π?

  1. π
  2. 4π
  3. 2π
Show answer

π (1/4) × 4π = π.

Check 8

What is the area of a sector with angle 120° and radius 3 in terms of π?

  1. 3π
  2. 9π
  3. 6π
Show answer

3π (1/3) × 9π = 3π.

Check 9

What is the length of an arc with angle 90° and radius 4 in terms of π?

  1. 2π
  2. 8π
  3. 4π
Show answer

2π (1/4) × 8π = 2π.

Check 10

What is the length of an arc with angle 60° and radius 3 in terms of π?

  1. π
  2. 6π
  3. 2π
Show answer

π (1/6) × 6π = π.

Check 11

What additional lengths must be added to the arc to find sector perimeter?

  1. Two straight radii (2r)
  2. One radius (r)
  3. The diameter (2r)
Show answer

Two straight radii (2r) A sector is enclosed by the curved arc and two straight radii.

Check 12

What is the perimeter of a sector with angle 90° and radius 4?

  1. 8 + 2π
  2. 2π
  3. 4 + 2π
Show answer

8 + 2π Arc is 2π; two radii are 4 + 4 = 8, so 8 + 2π.

Check 13

What is the perimeter of a sector with angle 45° and radius 4?

  1. 8 + π
  2. π
  3. 4 + π
Show answer

8 + π Arc is (45/360) × 8π = π; radii are 4 + 4 = 8, so 8 + π.

Check 14

What is the area of a semicircle (angle 180°) of radius 2 in terms of π?

  1. 2π
  2. 4π
  3. π
Show answer

2π (1/2) × 4π = 2π.

Check 15

Can you add the numerical radii to the coefficient of π in perimeter?

  1. No, leave as 8 + 2π
  2. Yes, write 10π
  3. Yes, write 10
Show answer

No, leave as 8 + 2π 8 and 2π are unlike terms; write as an exact sum.

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