Loading…
Loading…
Volume and surface area of every 3D shape on the IGCSE Edexcel specification. Formulae, composite-solid strategies, and worked examples for grades 5-8.
3D mensuration covers the volume and surface area of solid shapes including prisms, cylinders, pyramids, cones, spheres, and composite solids. You need to select the correct formula, substitute dimensions accurately, and handle multi-step problems where shapes are combined or subtracted. This topic appears on both Paper 1 and Paper 2 and typically carries 6-10 marks per exam sitting.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A prism is any 3D shape with a uniform cross-section. Its volume is found by multiplying the area of the cross-section by the length (or depth) of the prism.
Volume of a prism
V = cross-section area x length
A cylinder is a special prism whose cross-section is a circle. This gives us:
| Formula | Expression |
|---|---|
| Volume of cylinder | V = pi r squared h |
| Surface area of cylinder | SA = 2 pi r h + 2 pi r squared |
The surface area of a cylinder consists of two circular ends (each with area pi r squared) and the curved surface (a rectangle that wraps around, with area 2 pi r h when unrolled).
A pyramid has a polygonal base and triangular faces that meet at a single apex. The volume of any pyramid is one-third of the product of its base area and its perpendicular height.
Volume of a pyramid
V = (1/3) x base area x perpendicular height
For a square-based pyramid with side length s and height h, the base area is s squared, so the volume is (1/3) x s squared x h. The perpendicular height must be measured at right angles from the base to the apex, not along a sloping edge.
A cone can be thought of as a pyramid with a circular base. Its volume formula is therefore one-third of the base area (pi r squared) multiplied by the perpendicular height.
| Formula | Expression |
|---|---|
| Volume of cone | V = (1/3) pi r squared h |
| Curved surface area of cone | CSA = pi r l (where l = slant height) |
| Total surface area of cone | TSA = pi r l + pi r squared |
The slant height l can be found from the radius r and perpendicular height h using Pythagoras: l squared = r squared + h squared. If the question gives you the slant height but asks for the volume, you will need to calculate h first.
A sphere is the set of all points at a fixed distance (the radius r) from a centre point. The formulae for a sphere are provided on the IGCSE formula sheet, but you must be able to substitute and rearrange them confidently.
| Formula | Expression |
|---|---|
| Volume of sphere | V = (4/3) pi r cubed |
| Surface area of sphere | SA = 4 pi r squared |
A hemisphere is exactly half a sphere. Its volume is (2/3) pi r cubed, and its total surface area is the curved part (2 pi r squared) plus the flat circular face (pi r squared), giving 3 pi r squared in total.
Composite solids are made up of two or more standard 3D shapes joined together (or one shape removed from another). The strategy is always the same:
A common composite solid in IGCSE exams is a hemisphere sitting on top of a cylinder. The total volume is the cylinder volume plus the hemisphere volume. The total surface area is the curved surface of the cylinder, plus one circular base, plus the curved surface of the hemisphere (the flat face where they join is hidden).
A cylinder has radius 4 cm and height 10 cm. Find its volume and total surface area. Give your answers correct to 3 significant figures.
Volume:
V = pi r squared h = pi x 4 squared x 10 = pi x 16 x 10 = 160 pi
V = 502.654... = 503 cm cubed (3 s.f.)
Surface area:
SA = 2 pi r h + 2 pi r squared
= 2 pi (4)(10) + 2 pi (4 squared)
= 80 pi + 32 pi = 112 pi
= 351.858... = 352 cm squared (3 s.f.)
Answer: Volume = 503 cm cubed, Surface area = 352 cm squared
A cone has base radius 6 cm and slant height 10 cm. Calculate the volume of the cone. Give your answer correct to 3 significant figures.
Step 1 (Find perpendicular height using Pythagoras):
l squared = r squared + h squared
10 squared = 6 squared + h squared
100 = 36 + h squared
h squared = 64, so h = 8 cm
Step 2 (Calculate volume):
V = (1/3) pi r squared h = (1/3) pi (6 squared)(8) = (1/3) pi (36)(8) = 96 pi
V = 301.592... = 302 cm cubed (3 s.f.)
Answer: 302 cm cubed
A solid is formed by placing a hemisphere of radius 5 cm on top of a cylinder of the same radius and height 12 cm. Find the total volume and the total surface area of the solid. Give your answers correct to 3 significant figures.
Step 1 (Volume of cylinder):
V_cyl = pi r squared h = pi (25)(12) = 300 pi
Step 2 (Volume of hemisphere):
V_hemi = (1/2) x (4/3) pi r cubed = (2/3) pi (125) = (250/3) pi
Step 3 (Total volume):
V = 300 pi + (250/3) pi = (900/3 + 250/3) pi = (1150/3) pi
V = 1204.27... = 1200 cm cubed (3 s.f.)
Step 4 (Surface area):
Curved surface of cylinder = 2 pi r h = 2 pi (5)(12) = 120 pi
Bottom circle = pi r squared = 25 pi
Curved surface of hemisphere = 2 pi r squared = 50 pi
(The flat face where they join is hidden)
SA = 120 pi + 25 pi + 50 pi = 195 pi = 612.61... = 613 cm squared (3 s.f.)
Answer: Volume = 1200 cm cubed, Surface area = 613 cm squared
Using diameter instead of radius: Many questions give the diameter. Always halve it before substituting into a formula. This single error causes more lost marks in mensuration than any other.
Confusing slant height and perpendicular height: The volume formula for a cone requires the perpendicular height h, not the slant height l. If you are given the slant height, use Pythagoras to find h first.
Forgetting the 1/3 factor: The volume of a pyramid and cone both include a factor of 1/3. Forgetting this gives you three times the correct answer.
Counting hidden faces in surface area: When two shapes are joined (e.g. hemisphere on cylinder), the face where they meet is internal and must not be included in the total surface area.
Leave answers in terms of pi when asked: Some questions say "give your answer in terms of pi". This means do not evaluate pi as a decimal. Simply write your answer as a multiple of pi (e.g. 96 pi cm cubed).
Write the formula first: Examiners award a method mark for correctly stating the formula before substitution. Even if you make an arithmetic slip, the formula earns a mark.
Include correct units: Volume is always in cubic units (cm cubed, m cubed) and surface area in square units (cm squared, m squared). Missing or wrong units can lose a mark.
Use the ANS button for multi-step problems: On Paper 2, use your calculator's ANS function to carry full precision through each step. Only round at the very end to the required degree of accuracy.
Geometry & Trigonometry Hub
All geometry topics in one place
Worksheets & Answers
Free practice for geometry topics
Formula Sheet
Visual guide to key formulae
IGCSE Tutors in Dubai
Specialist in-home IGCSE support
Maths Tutors in Dubai
Expert maths tutoring at home
Contact GetYourTutors — IGCSE Maths 3D Mensuration
Phone: (+971) 4-313-2715 | Mobile: 050-947-9432
WhatsApp: 050-947-9432
Email: info@getyourtutors.com
Emirates Towers, Office Tower, Level 41, Sheikh Zayed Road, PO Box 31003, Dubai, UAE
Last updated: March 2026
Everything you need to know about our private tutoring services in Dubai.
A dedicated tutor builds confident volume and surface area skills from standard shapes through to composite solids. In-home across Dubai.