AQA GCSE Mathematics Paper 2 (Foundation), June 2025: Question 25

Circle Geometry: Circumference to Sector Area

📌 What this question tests

Rearranging circle formulas: Working backwards from circumference ( C = 2πr ) to find the radius.
Sector area calculation: Applying the sector formula ( Area = (θ/360) × πr² or using a quarter-circle fraction).
Rounding & precision: Maintaining accuracy with π throughout multi-step working.
Proportional reasoning: Understanding the relationship between angle size and sector area.

Question 25 (a)

Calculating the Shaded Sector Area [3 Marks]

Circumference = 20 cm, angle x = 90°

📐 Step-by-Step Method

Find the radius (r) from the circumference:
Formula: Circumference = 2 × π × r
Rearrange for r :
r = 20 ÷ (2π) = 10 / π ≈ 3.1831 cm
Awarded M1 for showing 20 ÷ 2π or obtaining a radius in the range [3.18, 3.185] (or diameter in [6.36, 6.37] ).
Calculate the area of the quarter circle (sector):
Since angle x = 90° , the fraction of the circle is 90° / 360° = 1/4 .
Sector Area = (90 / 360) × π × r²
Sector Area = 0.25 × π × (3.1831)² = 0.25 × π × 10.132 ≈ 7.9577 cm²
Alternative exact working: 1/4 × π × (10/π)² = 1/4 × π × (100/π²) = 25/π ≈ 7.9577 cm²
Awarded M1dep for complete correct method using their radius.
State final value as a decimal:
7.96 cm² (any value between 7.93 and 7.97 is accepted).
Awarded A1 for an accurate decimal answer in range.

✅ Mark Scheme Accepted Answers

Any decimal answer in the range:

7.93 to 7.97 cm²

(e.g., 7.96 or 7.958 ). If correct working is shown, subsequent rounding will not lose marks.

💡 Key Formulas to Memorise

Circumference = 2πr = πd
Circle Area = πr²
Sector Area = (θ / 360°) × πr²
A 90° angle at the centre always forms a quadrant (exactly 1/4 of the full circle).

❌ Common Traps & Misconceptions

Confusing diameter and radius: Writing r = 20 / π = 6.37 forgets to divide by 2! 6.37 cm is the diameter.
Premature rounding: Rounding r too early (e.g. to 3.2 ) gives 0.25 × π × 3.2² = 8.04 , which falls outside the acceptable mark range [7.93, 7.97] . Keep full values in calculator memory!
Leaving as a fraction with π: The question requests "as a decimal". Leaving the answer as 25/π will lose the final accuracy mark ( A1 ).

🧠 Examiner Advice & Technique

Use brackets on calculators: When calculating 20 ÷ 2π , type 20 ÷ (2 × π) . Typing 20 ÷ 2 × π will calculate (20/2) × π = 31.42 !
Show all working: Even if your final decimal slips slightly outside the range, writing 20 ÷ 2π guarantees the first method mark ( M1 ).

Question 25 (b)

Impact of Changing the Angle [1 Mark]

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

✅ Correct Option

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

Awarded B1 for selecting the first box only.

💡 The Mathematical Reason

The area of a sector is directly proportional to its central angle:

Area ∝ x

Since the radius of the circle has not changed, decreasing the angle x below 90° reduces the fraction x / 360° . A smaller fraction of the circle means a strictly smaller shaded area.