AQA A-Level Mathematics Paper 2, June 2025: Question 5

Mark scheme

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5 Recalls the formula for the area 1.2 M1 1 2

12 A = 8 1 4.

of a sector A = r 2

Or = 44 8. cm

Recalls A = r

and

changes their angle to

AWRT 80 2.o

224 1.1b A1

Obtains OE

Condone missing or incorrect

units

If units are converted the value

must be correct.

Question 5 Total 2

How to answer it

Area of a Circular Sector in Radians

AQA A-Level Mathematics • Pure Mathematics • Question 5 (2 Marks)

What this question tests

Recalling and applying the standard formula for the area of a sector when the angle is given in radians.
Correct mathematical substitution and accurate arithmetic computation without unnecessary conversions.
Understanding the relationship between radian measure and circle sector properties.

Question 5 Walkthrough

Sector with radius r = 8 cm and angle θ = 1.4 radians

💡 Key Knowledge

When an angle θ is measured in radians:

Arc Length: s = rθ
Sector Area: A = ½ r² θ

Note: The degree formula A = (θ / 360) × πr² is valid only when θ is in degrees. When using radians, the factor of 2π replaces 360°, simplifying directly to ½ r² θ .

📐 Step-by-Step Calculation

Identify given values:
Radius r = 8 cm
Angle θ = 1.4 rad
Substitute into radian area formula:
A = ½ × (8)² × 1.4
Evaluate the expression:
A = ½ × 64 × 1.4
A = 32 × 1.4 = 44.8 cm² (or 224/5 )

✅ Correct Answer & Mark Breakdown

Final Answer: 44.8 cm² or 224/5

M1 (AO 1.2): For recalling and substituting correctly into A = ½ r² θ , or for converting 1.4 rad ≈ 80.2° and using the degree formula.

A1 (AO 1.1b): For obtaining 44.8 or equivalent fraction ( 224/5 ). Examiner note: Missing or incorrect units are condoned, but if converted (e.g. to mm²), the numerical value must be correct.

🧠 Exam Technique & Examiner Insight

Work in radians directly: Never convert to degrees unless strictly required. Converting 1.4 rad into 80.21° introduces premature rounding errors and wastes valuable time.
Double-check formulas from the booklet: The formula A = ½ r² θ is standard Pure Mathematics knowledge. Make sure you don't confuse it with triangle area ½ r² sin θ .
Exact decimal vs fraction: Here, 44.8 is terminating and exact, so both decimal and fractional forms are awarded full credit immediately.

❌ Common Errors & Pitfalls

Confusing Sector with Segment / Triangle: Using ½ r² sin(θ) calculates the area of the triangle ΔAOB (approx. 31.5 cm² ), not the full curved sector.
Confusing Arc Length with Area: Using s = rθ = 8 × 1.4 = 11.2 gives arc length, scoring 0 marks.
Forgetting to square the radius: Evaluating ½ × 8 × 1.4 = 5.6 instead of ½ × 8² × 1.4 .
Unnecessary π factor: Multiplying by π inadvertently when applying the radian formula (e.g. thinking all circle formulas must contain π). In radians, π is already factored into the angle's definition.

Topics

Pure Mathematics · E: Trigonometry