Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Volume and surface area. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Volume and surface area of cuboids, prisms, cylinders, pyramids, cones and spheres, and the faces, edges and vertices of 3D shapes. Questions often add a context such as filling a tank or working out a cost. The cone and sphere formulas are given in the question when you need them.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
Count faces, edges and verticese.g. a cuboid has 6 faces, 12 edges and 8 vertices.
3
Find the volume of a cuboidVolume = length × width × height (or count the centimetre cubes).
4
Find volumes of prisms and cylindersVolume = area of cross-section × length; for a cylinder, V = πr²h.
4
Find the surface area of a prismAdd the areas of all the faces; sketch a net so you do not miss any.
5
Find the surface area of a cylinderTwo circles plus the curved surface: 2πr² + 2πrh.
5
Use formulas for pyramids, cones and spheresSubstitute into the formula, e.g. volume of a cone = ⅓πr²h.
Notes
3D shapes
Faces are the flat surfaces, edges are where two faces meet and vertices are the corners.
Find the area of the cross-section (the end face).
Volume: multiply by the length.
Surface area: add the two end faces and every rectangular face (each rectangle is one edge of the cross-section × the length).
Example. A prism has a cross-section that is a right-angled triangle with base 6 cm and height 8 cm. The prism is 20 cm long. (a) Work out the volume of the prism. (2 marks) (b) Work out the total surface area of the prism. (3 marks)
Show the model answer
(a) Cross-section = ½ × 6 × 8 = 24 cm²; volume = 24 × 20 (M1) = 480 cm³ (A1) (b) Third side of the triangle = √(6² + 8²) = 10 cm (M1) Area = 2 × 24 + 6 × 20 + 8 × 20 + 10 × 20 (M1) = 48 + 480 = 528 cm² (A1)
Cylinder in context (capacity and filling)
4 marks5
Work out the volume in cm³ (all lengths in cm).
Convert to litres: divide by 1000.
Use the rate or price given to answer the actual question.
Example. A water tank is a cylinder with radius 40 cm and height 90 cm. The empty tank is filled with water at a rate of 12 litres per minute. How long does it take to fill the tank? Give your answer to the nearest tenth of a minute.
The prism rule works for any prism: area of the end face × length.
Check faces, edges and vertices with F + V − E = 2 (cuboid: 6 + 8 − 12 = 2).
Look for Pythagorean triples in cones: 3-4-5 or 5-12-13 often link r, h and l.
A 10 cm cube holds 1000 cm³, which is exactly 1 litre; use this to sense check capacities.
Where marks are lost
Using the slant height instead of the perpendicular height in the volume of a cone or pyramid.
Forgetting the ⅓ for cones and pyramids.
Finding only the curved surface when the total surface area is asked for, or including faces that are joined together.
Mixing units, e.g. a radius in cm with a height in m.
Dividing by 100 instead of 1000 to change cm³ to litres.
Exam technique
Copy any formula given in the question and substitute your values into it clearly.
'In terms of π' means leave π in and keep fractions exact, e.g. 550π/3.
In filling or cost problems, make sure your final answer is what was asked (time, number of tins, cost), not just the volume.
Give units: cm³ for volume and cm² for surface area.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
The diagram shows a prism. The cross-section of the prism is a right-angled triangle. Diagram NOT accurately drawn
[object Object]
Work out the volume of the prism.[2]
Show the answer and mark scheme
Answer: 441 cm³
M1 for ½ × 7 × 9 (= 31.5) or ½ × 7 × 9 × 14
A1 for 441
Worked solution: Area of cross-section \(= \frac{1}{2} \times 7 \times 9 = 31.5\) cm² Volume \(= 31.5 \times 14 = 441\) cm³
Question 2Medium3 marks
A tank in the shape of a cuboid measures 40 cm by 30 cm by 25 cm. It is full of water. All of the water is poured into an empty cylinder with radius 15 cm and height 40 cm.
Will the water overflow? You must show your working.[3]
Show the answer and mark scheme
Answer: Yes: the water has volume 30 000 cm³ but the cylinder holds only \(\pi \times 15^2 \times 40 = 28\,274\) cm³.
M1 for \(40 \times 30 \times 25\) (= 30 000)
M1 for \(\pi \times 15^2 \times 40\) (= 28 274...)
C1 for 'yes', with 30 000 compared with 28 274 (or 28 300)
Worked solution: Water: \(40 \times 30 \times 25 = 30\,000\) cm3. Cylinder: \(\pi \times 225 \times 40 = 28\,274.3\ldots\) cm3. 30 000 > 28 274, so the water overflows.