AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 15

Sphere Radius and Volume in Terms of π

📋 What this question tests

Relationship between diameter and radius: Understanding that radius = diameter ÷ 2 .
Mathematical justification: Writing a clear arithmetic or verbal reason to "show that" a given result is correct.
Substituting into a geometric formula: Evaluating the volume of a sphere V = (4/3)πr³ .
Exact answers: Giving answers left in exact terms of π as a simplified fraction.

Question 15 (a) • 1 Mark

Show that the radius of the sphere is 10 cm

Relationship between diameter and radius

✅ Model Answer

Calculation: 20 ÷ 2 = 10

Or in words: "The radius is half of the diameter" or "10 × 2 = 20".

Mark allocation: [1 mark total] B1 for showing division by 2 or stating that radius is half of diameter.

💡 Key Knowledge

Diameter (d): The full distance across the sphere through its centre.
Radius (r): The distance from the centre to the edge.
Formula: r = d ÷ 2 or d = 2r .

🧠 Exam Technique: "Show that" questions

When an exam question says "Show that the radius is 10 cm", the answer 10 is already given to you! You cannot simply write "10 cm". You must write the mathematical operation that leads to 10.

❌ Common Errors in Part (a)

Writing 20 - 10 = 10 (Scores 0 marks because radius is found by division/halving, not subtraction).
Vague qualitative statements like "Radius is smaller than diameter" (Scores 0 marks).
Attempting to use volume formulas to work backwards unnecessarily.

Question 15 (b) • 2 Marks

Calculate the volume in terms of π

Using the volume of a sphere formula: V = (4/3)πr³

📐 Step-by-Step Calculation

Identify the radius:
Diameter = 20 cm, so radius r = 10 cm (from part a).
Cube the radius:
r³ = 10³ = 10 × 10 × 10 = 1000
(Awards M1 mark)
Multiply by (4/3)π:
Volume = (4/3) × π × 1000 = (4000/3)π cm³

✅ Acceptable Final Answers

Exact fraction form (preferred):

(4000/3)π or 4000π / 3

Also accepted:

1333.3...π or 1333⅓ π
Condone explicit multiplication signs, e.g. (4000/3) × π

Mark allocation:
• M1: For calculating 10³ or 1000 .
• A1: For (4000/3)π or equivalent exact value.

❌ Common Traps to Avoid

Using the diameter instead of the radius: Substituting r = 20 gives (4/3) × π × 20³ = (32000/3)π . The mark scheme awards a Special Case score of SC1 for this error. Always double-check whether the question provides diameter or radius!
Evaluating π on the calculator: The question states "Give your answer in terms of π". If you multiply out π into a decimal like 4188.79 , you lose the final accuracy mark.
Squaring instead of cubing: Mistakenly calculating 10² = 100 instead of 10³ = 1000 . Sphere volume requires r³ .
Incomplete simplification: Leaving the answer written as 4000π ÷ 3 without writing it as a single fraction may fail to secure full marks (scores M1A0).

🧠 Examiner Insights & Top Tips

Notice part (a) was a hint: Part (a) deliberately guides you to find r = 10 so that you don't accidentally plug 20 into the formula in part (b).
Units are already provided: Look at the answer line: cm³ is already printed for you, so focus purely on getting the numerical coefficient and π correct.
Keep it in fraction form: Since 4000 ÷ 3 is a recurring decimal, the cleanest and most accurate way to present your answer is as the improper fraction 4000/3 π .