📏
Geometry & Measures · Higher

Volume of pyramids, cones & spheres

Pyramids, cones and spheres have specific volume formulas. These are given on the formula sheet in some exams, but you need to know how to apply them correctly.

🔑

Key facts to remember

  • 1Volume of a pyramid = ⅓ × base area × height.
  • 2Volume of a cone = ⅓ × π × r² × h.
  • 3Volume of a sphere = (4/3) × π × r³.
  • 4Curved surface area of a cone = πrl (where l = slant height).
  • 5Surface area of a sphere = 4πr².
  • 6The height used in the cone formula is the perpendicular height, not the slant height.
📐

Formulas

Pyramid volume
V = ⅓ × base area × h
Cone volume
V = ⅓πr²h
Sphere volume
V = (4/3)πr³
Cone curved surface area
CSA = πrl

l = slant height

Sphere surface area
SA = 4πr²
✍️

Worked examples

Example 1

Find the volume of a cone with radius 5 cm and perpendicular height 9 cm. Give your answer in terms of π.

Working

  1. V = ⅓ × π × r² × h
  2. V = ⅓ × π × 25 × 9
  3. V = ⅓ × 225π = 75π cm³
Answer75π cm³
Example 2

A sphere has a volume of 288π cm³. Find its radius.

Working

  1. (4/3)πr³ = 288π
  2. (4/3)r³ = 288
  3. r³ = 288 × (3/4) = 216
  4. r = ∛216 = 6 cm
Answerr = 6 cm
⚠️

Common mistakes

✗Using the slant height instead of the perpendicular height for cone volume.
✗Forgetting the ⅓ in the pyramid and cone formulas.
✗Squaring instead of cubing r in the sphere volume formula.
🎯

Exam tips

✓These formulas may be given — but know what each letter represents so you can substitute correctly.
✓For spheres and cones: identify r first; for cones also identify h (perpendicular) and l (slant) separately.

Ready to test yourself on Volume of pyramids, cones & spheres?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →
← All topicsDashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

▶volume cone sphere pyramid GCSE Higher▶sphere volume formula GCSE maths▶cone volume GCSE Higher▶pyramid volume GCSE maths

Opens YouTube — pick any free GCSE video.