Volume of a Hemisphere –

Volume of a hemisphere is a Higher-tier GCSE Maths topic tested on AQA, Edexcel, and OCR papers. A hemisphere is exactly half of a sphere, so you halve the sphere volume formula. Questions may also ask for the total surface area, which includes the curved surface and the flat circular base. This guide explains both formulas, works through exam-style problems, and gives you practice questions.

What Is a Hemisphere?

A hemisphere is half of a sphere, created by cutting a sphere along a great circle (a circle passing through the centre). It has a curved surface (half the sphere) and a flat circular base.

Key Formulas

The total surface area includes the curved part (half the sphere's surface area of 4πr²) plus the flat circular base (πr²).

Step-by-Step Method

Finding the Volume

1. Identify the radius of the hemisphere.
2. Substitute into V = (2/3)πr³.
3. Evaluate and round as required.

Finding the Total Surface Area

1. Calculate the curved surface area: 2πr².
2. Calculate the flat base area: πr².
3. Add them together: total SA = 3πr².

Worked Example 1 — Foundation Level

This topic is Higher only, but this example uses straightforward numbers.

Question: Find the volume of a hemisphere with radius 6 cm. Give your answer in terms of π.

Working:

Step 1 — V = (2/3)πr³.

Step 2 — V = (2/3) × π × 6³ = (2/3) × π × 216.

Step 3 — V = (2 × 216 / 3) × π = (432/3) × π = 144π cm³.

Answer: The volume is 144π cm³.

Worked Example 2 — Higher Level

Question: A hemisphere has a volume of 486π cm³. Find the radius.

Working:

Step 1 — Use V = (2/3)πr³. Substitute: 486π = (2/3)πr³.

Step 2 — Divide both sides by π: 486 = (2/3)r³.

Step 3 — Multiply both sides by 3/2: r³ = 486 × 3/2 = 729.

Step 4 — Cube root: r = ∛729 = 9 cm.

Answer: The radius is 9 cm.

Worked Example 3 — Exam Style

Question: A solid shape consists of a cone of height 12 cm mounted on top of a hemisphere. Both share a circular base of radius 5 cm. Find the total volume of the shape to 1 decimal place.

Working:

Step 1 — Volume of hemisphere = (2/3)πr³ = (2/3) × π × 125 = (250/3)π cm³.

Step 2 — Volume of cone = (1/3)πr²h = (1/3) × π × 25 × 12 = 100π cm³.

Step 3 — Total volume = (250/3)π + 100π = (250/3 + 100)π = (250/3 + 300/3)π = (550/3)π.

Step 4 — (550/3)π = 183.333... × 3.14159... = 575.958... ≈ 576.0 cm³.

Answer: The total volume is 576.0 cm³.

Common Mistakes

• Using the full sphere formula instead of halving. Always divide the sphere volume by 2: V = (2/3)πr³, not (4/3)πr³.
• Forgetting the flat base in surface area. The total surface area of a hemisphere is 3πr², not just the curved part (2πr²). Read the question carefully to see if it asks for curved SA or total SA.
• Using diameter instead of radius. If given the diameter, halve it before substituting into the formula.

Exam Tips

• Formulas for sphere volume and surface area are given on the formula sheet — but you must remember to halve them for a hemisphere.
• Leave your answer in terms of π if the question says "give your answer in terms of π" — do not convert to a decimal.
• Composite shape questions (hemisphere + cone, hemisphere + cylinder) are common — calculate each part separately and add.
• When working backwards to find r, remember to take the cube root at the end.

Practice Questions

Q1 (Higher): Find the volume of a hemisphere with radius 10 cm. Give your answer to 1 decimal place.

Q2 (Higher): Find the total surface area of a hemisphere with radius 7 cm. Give your answer in terms of π.

Q3 (Higher): A hemisphere has a total surface area of 75π cm². Find the radius.

Practise hemisphere volume and surface area questions with instant feedback free on GCSEMathsAI.

Related Topics

• Volume of 3D Shapes — volumes of prisms, pyramids, cones, and spheres.
• Surface Area — surface area of various 3D shapes.
• Arc Length and Sector Area — working with parts of circles.

Summary

A hemisphere is half a sphere with volume (2/3)πr³ and total surface area 3πr². The most frequent errors are forgetting to halve the sphere formula and omitting the flat circular base from the surface area. On Higher papers, hemisphere questions often appear as part of composite shapes combined with cones or cylinders — calculate each part separately and sum the results. Always check whether the question asks for exact answers in terms of π or decimal approximations.