GCSE Maths: Arc lengths and areas of sectors (Foundation)
Part of the GCSE Maths perimeter, area and circles chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Arc lengths and areas of sectors
A sector is a slice of a circle between two radii, like a slice of pizza. Its curved edge is an arc. The angle θ between the two radii (at the centre) tells you what fraction of the whole circle the slice is.
A full turn is 360°. So a sector with angle 90° is 90 out of 360, which is ¼ of the circle. A sector with angle 60° is 60/360 = ⅙ of the circle. In general the fraction is θ ÷ 360 (simplify it if you can). Because the slice is that fraction of the circle, both its arc length and its area are the same fraction of the whole circle’s values:
Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr2
Notice what is the same and what is different. Both have the fraction θ/360 in front. The arc length starts from the circumference 2πr (a length), the area starts from πr2 (an area). The perimeter of a sector is the arc plus the two straight radii: arc + 2r.
Working backwards
If you know the arc length or area, write the formula as an equation and solve for the angle θ or the radius r. For example: if the arc length is 4π cm and r = 12 cm, then 4π = (θ/360) × 2π × 12. Divide by π and simplify to get θ.
Key ideas
- Fraction of the circle = θ ÷ 360 (simplify it).
- Arc length = (θ/360) × 2πr.
- Sector area = (θ/360) × πr2.
- Perimeter of a sector = arc length + 2 × radius.
- Use the same fraction for the arc and the area. Do not mix up 2πr (arc) and πr2 (area).
- To find θ or r: write the formula as an equation and undo it step by step.
Watch out
The perimeter of a sector is not just the arc. Do not forget the two straight edges (the radii). Also check that you divide the angle by 360, not by 180.Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.60360 = 16
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 6 = 12π
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 16 × 12π
- Simplify and keep π. Multiply the number parts together; leave π.arc length = 2π cm
Check. As a decimal: 2 × 3.14 = 6.28 cm. Compare with the circumference 37.68 cm: the arc is 16 of it. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.120360 = 13
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 92 = 81π
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 13 × 81π
- Simplify and keep π. Multiply the number parts together.sector area = 27π cm²
Check. The whole circle has area 81π. The sector is 13 of it, and 13 × 81 = 27. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.75360 = 524
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 8 = 50.2655…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 524 × 50.2655…
- Round to 1 decimal place. Look at the next digit.arc length = 10.5 cm
Check. The arc should be 524 of the full circumference 50.27 cm: 524 × 50.27 = 10.47. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The perimeter of a sector is the curved arc plus the two straight radii. Do not forget the two radii.
- Arc length. The arc is 16 of the circumference 2πr.16 × 2 × π × 6 = 2π cm
- Two radii. The two straight edges are both radii.2 × 6 = 12 cm
- Combine the pieces. Add. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Perimeter = 2π + 12 = (12 + 2π) cm
Check. Estimate with π ≈ 3: the answer should be about 18 cm, which is close to the value found. ✓
Watch out. A sector has three edges: the arc and two radii. Adding only the arc is a very common slip.
Method. The arc is θ/360 of the circumference. Write that as an equation and solve for θ.
- Write the equation. arc length = (θ ÷ 360) × 2πr.4π = θ360 × 2 × π × 12
- Simplify the right-hand side. Multiply the numbers 2 × r.4π = θ360 × 24π
- Divide both sides by π. The π’s cancel.4 = θ360 × 24
- Divide by 24. Undo the multiplication.16 = θ360
- Multiply by 360. Undo the division by 360.θ = 16 × 360 = 60°
Check. 60 ÷ 360 = 16, and 16 of the circumference 24π is 4π. ✓
Watch out. Divide by π at the start so the π’s cancel. Then θ is found by multiplying by 360.
Exercise 18H Arc lengths and areas of sectors
Calculator allowedMethod. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
The highest common factor is 72. - Divide the top and the bottom by 72. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.72 ÷ 72 = 1 and 360 ÷ 72 = 5
72360 = 15
Check. 72360 = 0.2 as a decimal, and 15 = 0.2 too, so the value has not changed.
Watch out. Divide top and bottom. If you only divide one of them you change the value of the fraction.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.90360 = 14
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 4 = 8π
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 14 × 8π
- Simplify and keep π. Multiply the number parts together; leave π.arc length = 2π cm
Check. As a decimal: 2 × 3.14 = 6.28 cm. Compare with the circumference 25.12 cm: the arc is 14 of it. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.60360 = 16
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 6 = 12π
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 16 × 12π
- Simplify and keep π. Multiply the number parts together; leave π.arc length = 2π cm
Check. As a decimal: 2 × 3.14 = 6.28 cm. Compare with the circumference 37.68 cm: the arc is 16 of it. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.60360 = 16
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 62 = 36π
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 16 × 36π
- Simplify and keep π. Multiply the number parts together.sector area = 6π cm²
Check. The whole circle has area 36π. The sector is 16 of it, and 16 × 36 = 6. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.90360 = 14
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 10 = 62.8319…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 14 × 62.8319…
- Round to 1 decimal place. Look at the next digit.arc length = 15.7 cm
Check. The arc should be 14 of the full circumference 62.83 cm: 14 × 62.83 = 15.71. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.90360 = 14
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 102 = 314.1593…
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 14 × 314.1593…
- Round to 1 decimal place. Look at the next digit.sector area = 78.5 cm²
Check. The sector should be 14 of the whole circle area 314.16 cm²: 78.54. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.75360 = 524
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 8 = 50.2655…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 524 × 50.2655…
- Round to 1 decimal place. Look at the next digit.arc length = 10.5 cm
Check. The arc should be 524 of the full circumference 50.27 cm: 524 × 50.27 = 10.47. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.75360 = 524
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 82 = 201.0619…
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 524 × 201.0619…
- Round to 1 decimal place. Look at the next digit.sector area = 41.9 cm²
Check. The sector should be 524 of the whole circle area 201.06 cm²: 41.89. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.120360 = 13
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 9 = 18π
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 13 × 18π
- Simplify and keep π. Multiply the number parts together; leave π.arc length = 6π cm
Check. As a decimal: 6 × 3.14 = 18.84 cm. Compare with the circumference 56.52 cm: the arc is 13 of it. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.120360 = 13
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 92 = 81π
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 13 × 81π
- Simplify and keep π. Multiply the number parts together.sector area = 27π cm²
Check. The whole circle has area 81π. The sector is 13 of it, and 13 × 81 = 27. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Method. The perimeter of a sector is the curved arc plus the two straight radii. Do not forget the two radii.
- Arc length. The arc is 16 of the circumference 2πr.16 × 2 × π × 6 = 2π cm
- Two radii. The two straight edges are both radii.2 × 6 = 12 cm
- Combine the pieces. Add. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Perimeter = 2π + 12 = (12 + 2π) cm
Check. Estimate with π ≈ 3: the answer should be about 18 cm, which is close to the value found. ✓
Watch out. A sector has three edges: the arc and two radii. Adding only the arc is a very common slip.
Method. The perimeter of a sector is the curved arc plus the two straight radii. Do not forget the two radii.
- Arc length. The arc is 16 of the circumference 2πr.16 × 2 × π × 6 = 6.2832… cm
- Two radii. The two straight edges are both radii.2 × 6 = 12 cm
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Perimeter = 6.2832… + 12 = 18.2832… cm
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Perimeter = 18.3 cm
Check. Estimate with π ≈ 3: the answer should be about 18 cm, which is close to the value found. ✓
Watch out. A sector has three edges: the arc and two radii. Adding only the arc is a very common slip.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.250360 = 2536
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 5 = 31.4159…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 2536 × 31.4159…
- Round to 1 decimal place. Look at the next digit.arc length = 21.8 cm
Check. The arc should be 2536 of the full circumference 31.42 cm: 2536 × 31.42 = 21.82. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.250360 = 2536
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 52 = 78.5398…
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 2536 × 78.5398…
- Round to 1 decimal place. Look at the next digit.sector area = 54.5 cm²
Check. The sector should be 2536 of the whole circle area 78.54 cm²: 54.54. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.45360 = 18
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 152 = 706.8583…
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 18 × 706.8583…
- Round to 1 decimal place. Look at the next digit.sector area = 88.4 cm²
Check. The sector should be 18 of the whole circle area 706.86 cm²: 88.36. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.45360 = 18
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 15 = 94.2478…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 18 × 94.2478…
- Round to 1 decimal place. Look at the next digit.arc length = 11.8 cm
Check. The arc should be 18 of the full circumference 94.25 cm: 18 × 94.25 = 11.78. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).
Method. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.140360 = 718
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 82 = 201.0619…
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 718 × 201.0619…
- Round to 1 decimal place. Look at the next digit.sector area = 78.2 m²
Check. The sector should be 718 of the whole circle area 201.06 m²: 78.19. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Method. The fence goes along the arc only. Find the arc length, then multiply by the cost per metre.
- Fraction of the circle. 100 out of 360.100360 = 518
- Arc length. Fraction × circumference 2πr.518 × 2 × π × 12 = 20.944… m
- Cost. Multiply by £8 per metre (do not round first).20.944… × 8 = £167.55…
- Round to the nearest pound. £168
Check. Estimate: the whole circumference is about 75 m; 5/18 of 75 is about 21 m; 21 × 8 = 168. ✓
Watch out. The fence is only along the curved edge, so use the arc length, not the perimeter of the sector.
Method. The arc is θ/360 of the circumference. Write that as an equation and solve for θ.
- Write the equation. arc length = (θ ÷ 360) × 2πr.4π = θ360 × 2 × π × 12
- Simplify the right-hand side. Multiply the numbers 2 × r.4π = θ360 × 24π
- Divide both sides by π. The π’s cancel.4 = θ360 × 24
- Divide by 24. Undo the multiplication.16 = θ360
- Multiply by 360. Undo the division by 360.θ = 16 × 360 = 60°
Check. 60 ÷ 360 = 16, and 16 of the circumference 24π is 4π. ✓
Watch out. Divide by π at the start so the π’s cancel. Then θ is found by multiplying by 360.
Method. Work out the arc length and the area of both sectors in terms of π and compare.
- Arc length of A. (40/360) × 2π × 9 = &frac19; × 18π.2π cm
- Arc length of B. (80/360) = &frac29; of 18π.4π cm
- Area of A. &frac19; × π × 81.9π cm²
- Area of B. &frac29; × π × 81.18π cm²
- Compare. Divide B by A.4π ÷ 2π = 2 and 18π ÷ 9π = 2, so B is double in both.
Check. The angle of B is double the angle of A, so it is double the fraction of the circle. ✓
Watch out. Doubling the angle doubles the fraction, so doubles both the arc length and the area.
Method. Compare her working with the arc length formula.
- Spot the mistake. πr2 is the area of a circle. An arc length is a distance, so it needs the circumference 2πr.Ella found the area of the sector (6π cm²), not the arc length.
- Correct it. Arc length = (θ/360) × 2πr.⅙ × 2 × π × 6 = ⅙ × 12π = 2π cm
Check. The full circumference is 12π, and 60° is ⅙ of a turn: 12π ÷ 6 = 2π. ✓
Watch out. Arc length uses 2πr; sector area uses πr2.