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Learn8300Papers 1–3 · Higher tierGeometry and measuresPrisms and cylinders
Geometry and measures · Subtopic 8 of 17

Prisms and Cylinders

A prism has the same cross-section all the way through, so its volume is cross-section area ×\times length. A cylinder is a prism with a circular cross-section. Surface area is the total area of every face: unfold the solid into a net and add.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Find the volume of cuboids, prisms and cylinders.
  • Find surface area by adding the areas of every face.
  • Convert volumes to litres.
  • Solve problems involving filling and pouring.
Pause and think

Slice a loaf of bread. Every slice has the same shape. How would you find the volume of the whole loaf from one slice?

Section 2 of 14

Core idea

Start hereA loaf of bread

Slice a loaf anywhere and every slice has the same shape. Solids like this are prisms, and their volume is simply the area of one slice times the length of the loaf.

The big idea

Prism volume = cross-section area × length. Cylinder: V=πr2hV=\pi r^2h, SA=2πr2+2πrhSA=2\pi r^2+2\pi rh. 1 cm³ = 1 ml.

1. Volume

A triangular prism: slices of the same triangle
Alength l
V=A×lV=A\times l. A stack of identical triangular slices.
Worked exampleTriangular prism

Triangle base 6, height 4; length 10.

Goal
Volume.
Strategy
Cross-section × length.
  1. 12(6)(4)=12\tfrac12(6)(4)=12
    Cross-section.
  2. 12×10=12012\times10=120
    Volume.
Answer120120 cm³.
✓ CheckHalf of a 6×4×106\times4\times10 box (240240) ✓.

2. Cylinders

The net of a cylinder
curved surface: 2πr × hwidth = circumference 2πrh
The curved surface unrolls into a rectangle whose width is the circumference of the end circle.
Surface area from the net
  1. 2×πr22\times\pi r^2
    Top and bottom circles.
  2. 2πr×h2\pi r\times h
    Curved surface unrolls to a rectangle: width = circumference.
SA=2πr2+2πrhSA=2\pi r^2+2\pi rh.
Worked exampler=3r=3, h=10h=10

Volume, surface, capacity.

Goal
Values.
Strategy
Formulas.
  1. V=π(9)(10)=90π=282.7V=\pi(9)(10)=90\pi=282.7
    Volume.
  2. SA=18π+60π=78π=245.0SA=18\pi+60\pi=78\pi=245.0
    Surface.
  3. 282.7 ml282.7\text{ ml}
    1 cm³ = 1 ml.
Answer282.7282.7 cm³; 245.0245.0 cm²; 282.7282.7 ml.
✓ Check90π≈28390\pi\approx283 ✓.
Common misconceptionMixed units Convert all lengths to the same unit before multiplying: 11 m³ =1 000 000=1\,000\,000 cm³.
Pause and think

Cuboid 2 m × 50 cm × 30 cm in m³?

Summary

  • Prism: area × length.
  • Cylinder from its net.
Key words
Prism
Constant cross-section solid.
HistoryEureka! The story goes that Archimedes noticed the water level rise as he got into his bath, and realised he could measure the volume of an irregular object (a king's crown) by the water it pushed aside. He then ran through the streets shouting "Eureka!" ("I have found it!").
Fun factAn Olympic pool in litres. An Olympic pool is at least 5050 m by 2525 m by 22 m: 25002500 m3^3, or 2.52.5 million litres. At two litres a day, that is more than 30003000 years of drinking water for one person.
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