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Spheres & cones practice questions with answers

Foundation and Higher · Grade 5 · Calculator

A second set of spheres & cones questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

Work out the volume of a sphere of radius 6. Give your answer in terms of π.

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Answer: 288π

  1. Cube radius: 6^3 = 216.
  2. Multiply by 4/3: (4/3) × 216π = 288π.
  • Method (1): Cube radius: 6^3 = 216.
  • Accuracy (1): Multiply by 4/3: (4/3) × 216π = 288π.

Question 2 Easy

Work out the volume of a sphere of radius 6. Give your answer in terms of π.

Show answer

Answer: 288π

  1. Cube radius: 6^3 = 216.
  2. Multiply by 4/3: (4/3) × 216π = 288π.
  • Method (1): Cube radius: 6^3 = 216.
  • Accuracy (1): Multiply by 4/3: (4/3) × 216π = 288π.

Question 3 Medium

Work out the volume of a hemisphere of radius 6. Give your answer in terms of π.

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Answer: 144π

  1. Full sphere volume: 288π.
  2. Half sphere volume: 288π / 2 = 144π.
  • Method (1): Full sphere volume: 288π.
  • Accuracy (1): Half sphere volume: 288π / 2 = 144π.

Question 4 Medium

Work out the volume of a hemisphere of radius 2. Give your answer in terms of π.

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Answer: 16π/3

  1. Full sphere volume: 32π/3.
  2. Half of 32π/3 is 16π/3.
  • Method (1): Full sphere volume: 32π/3.
  • Accuracy (1): Half of 32π/3 is 16π/3.

Question 5 Hard

Work out the volume of a cone of radius 2 and height 3. Give your answer in terms of π.

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Answer: 4π

  1. Base area times height: 4 × 3 = 12.
  2. Divide by 3: 12π / 3 = 4π.
  • Method (1): Base area times height: 4 × 3 = 12.
  • Accuracy (1): Divide by 3: 12π / 3 = 4π.

Question 6 Hard

Work out the surface area of a sphere of radius 1. Give your answer in terms of π.

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Answer: 4π

  1. Square radius: 1^2 = 1.
  2. Formula 4πr^2: 4 × π × 1^2 = 4π.
  • Method (1): Square radius: 1^2 = 1.
  • Accuracy (1): Formula 4πr^2: 4 × π × 1^2 = 4π.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Work out the volume of a cone of radius 3 and height 6. Give your answer in terms of π.

Show answer

Answer: 18π

  1. Base area is π × 3^2 = 9π.
  2. Multiply by height and divide by 3: 9π × 6 / 3 = 18π.
  • Method (1): (1/3) × π × 3^2 × 6
  • Accuracy (1): 18π

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What is the formula for the volume of a sphere of radius r?

  1. V = (4/3)πr^3
  2. V = 4πr^2
  3. V = (1/3)πr^2 h
Show answer

V = (4/3)πr^3 Volume cubes radius and multiplies by 4/3.

Check 2

What is the formula for the surface area of a sphere of radius r?

  1. A = 4πr^2
  2. A = (4/3)πr^3
  3. A = 2πr
Show answer

A = 4πr^2 Surface area is four circular cross-sections.

Check 3

What is the formula for the volume of a cone of radius r and height h?

  1. V = (1/3)πr^2 h
  2. V = πr^2 h
  3. V = (4/3)πr^3
Show answer

V = (1/3)πr^2 h A cone is one third of a cylinder with the same base and height.

Check 4

What is the volume of a sphere of radius 3 in terms of π?

  1. 36π
  2. 12π
  3. 9π
Show answer

36π (4/3) × 27π = 36π.

Check 5

What is the volume of a sphere of radius 2 in terms of π?

  1. 32π/3
  2. 16π
  3. 8π/3
Show answer

32π/3 (4/3) × 8π = 32π/3.

Check 6

What is the surface area of a sphere of radius 3 in terms of π?

  1. 36π
  2. 12π
  3. 18π
Show answer

36π 4 × 3^2 π = 4 × 9π = 36π.

Check 7

What is the surface area of a sphere of radius 2 in terms of π?

  1. 16π
  2. 8π
  3. 32π/3
Show answer

16π 4 × 2^2 π = 4 × 4π = 16π.

Check 8

What is the volume of a hemisphere of radius 3 in terms of π?

  1. 18π
  2. 36π
  3. 9π
Show answer

18π Half of 36π is 18π.

Check 9

What is the volume of a hemisphere of radius 2 in terms of π?

  1. 16π/3
  2. 32π/3
  3. 8π/3
Show answer

16π/3 Half of 32π/3 is 16π/3.

Check 10

What is the volume of a cone of radius 3 and height 4 in terms of π?

  1. 12π
  2. 36π
  3. 48π
Show answer

12π (1/3) × 9 × 4π = 12π.

Check 11

What is the volume of a cone of radius 3 and height 8 in terms of π?

  1. 24π
  2. 72π
  3. 18π
Show answer

24π (1/3) × 9 × 8π = 24π.

Check 12

What is the volume of a cone of radius 2 and height 3 in terms of π?

  1. 4π
  2. 12π
  3. 6π
Show answer

4π (1/3) × 4 × 3π = 4π.

Check 13

Why must you cube radius for sphere volume but square for surface area?

  1. Volume has 3 dimensions; surface area has 2 dimensions
  2. Because π is cubed in volume
  3. It is an arbitrary definition
Show answer

Volume has 3 dimensions; surface area has 2 dimensions Spatial volume scales as length cubed; area scales as length squared.

Check 14

What is the surface area of a sphere of radius 5 in terms of π?

  1. 100π
  2. 25π
  3. 500π/3
Show answer

100π 4 × 5^2 π = 4 × 25π = 100π.

Check 15

What does leaving an answer in terms of π mean?

  1. Keep the symbol π without approximating it as 3.14
  2. Divide the answer by π
  3. Multiply by 3.142
Show answer

Keep the symbol π without approximating it as 3.14 Express exact values using the symbol π.

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