AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 8

Circle Sectors: Circumference, Area & Angle Reasoning

What this question tests

This multi-step geometry problem assesses your ability to:

Rearrange the circumference formula ( C = 2πr or C = πd ) to determine the radius.
Calculate the area of a circle ( A = πr² ) and find a fractional sector area using angle proportion ( θ / 360° ).
Reason qualitatively about how changing an angle changes the resulting sector area.

Question 8 (a) • 3 Marks

Calculating the Shaded Sector Area

Given: Circumference = 20 cm, angle x = 90°

📐 Step-by-Step Calculation

1 Find the radius (r):
C = 2 × π × r = 20
r = 20 ÷ (2π) = 10 ÷ π ≈ 3.183 cm

2 Find the full circle area:
Area = π × r²
Area = π × (3.1831...)² ≈ 31.831 cm²

3 Calculate the sector area (90°):
A 90° angle represents a quarter of a full turn ( 90° / 360° = 1/4 ):
Shaded Area = 31.831 ÷ 4 ≈ 7.96 cm²

✅ Correct Answer & Mark Scheme

Acceptable Final Answer: Any decimal value in the range 7.93 to 7.97 (e.g. 7.96 cm²).

Mark Breakdown:
• [M1] For finding diameter ( 20 ÷ π , range [6.36, 6.4]) OR finding radius ( 20 ÷ 2π , range [3.18, 3.2]).
• [M1 dep] Complete method to find shaded area: π × (20 ÷ 2π)² × (90 ÷ 360) or equivalent.
• [A1] Accurate value in range [7.93, 7.97] .

💡 Key Knowledge

Circumference: C = πd = 2πr
Full Area: A = πr²
Sector Area: A = (x / 360) × πr²
Since x = 90° , the sector is simply a quadrant ( 1/4 of the circle).

❌ Common Calculation Traps

Calculator Error: Typing 20 / 2π without brackets gives (20 / 2) × π = 31.42 . Always type 20 ÷ (2 × π) or write as a fraction!
Radius vs. Diameter: Dividing 20 ÷ π gives diameter ( 6.37 ), not radius. Forgetting to divide by 2 leads to an answer 4 times too large.
Premature Rounding: Rounding radius to 3.2 too early gives π × 3.2² / 4 = 8.04 , which falls outside the acceptable mark scheme range [7.93, 7.97]. Keep values stored in your calculator!

Question 8 (b) • 1 Mark

Effect of Changing the Angle

"In fact, angle x is smaller than 90°. What does this mean about the shaded area?"

✅ Correct Selection

Tick the first box:

☑ It is smaller than the answer to part (a)

Mark Breakdown:
• [B1] Correct box checked.

🧠 Exam Technique & Reasoning

Sector area is directly proportional to the angle at the centre ( x ):

Area = (x / 360) × πr²

The radius and full circle area have not changed.
If x < 90° , then the fraction x / 360 < 90 / 360 .
Therefore, the shaded sector must cover a smaller proportion of the circle, making its area strictly smaller than part (a).