If you're studying for your GCSE Maths (AQA, Edexcel, or OCR), understanding how to find the volume of 3D shapes is essential. Two of the most common shapes you'll encounter are prisms and cylinders. This guide will walk you through exactly how to tackle these questions, step-by-step, with plenty of examples.
What is a Prism?
A prism is a 3D shape that has a constant cross-section along its length. This means if you were to slice through it parallel to one of its ends, the shape you get (the cross-section) is always the same. Common prisms include:
Cuboids (cross-section is a rectangle)
Triangular prisms (cross-section is a triangle)
- L-shaped prisms (cross-section is an L-shape)
What is a Cylinder?
A cylinder is technically a type of prism where the cross-section is a circle. You can use the prism formula for it, but it has its own specific version that you need to know.
The Golden Rule: The Volume Formula
For any prism, the volume is calculated using this fundamental formula:
Volume of a prism = Area of the cross-section × Length
Where:
Area of the cross-section: The area of the 2D shape that is repeated along the length of the prism.
- Length: The distance between the two identical cross-sections (sometimes called the height or depth).
For a cylinder, this formula becomes specific:
Volume of a cylinder = π × radius² × height or V = πr²h
Here, the cross-section is a circle (area = πr²), and the height is the length.
Your Step-by-Step Strategy
Follow this process every time you face a volume question. It turns a complex 3D problem into simpler 2D steps.
flowchart TD
A["Start: Identify the 3D Shape"] --> B["Is it a Prism, or Cylinder?"]
B -->|Prism| C["Step 1: Identify the, Constant Cross-Section"]
B -->|Cylinder| D["Use Formula: V = πr²h"]
C --> E["Step 2: Calculate the Area, of the Cross-Section"]
E --> F["Step 3: Identify the Length, of the Prism"]
F --> G["Step 4: Multiply:, Volume = Area × Length"]
D --> H["Step 5: State your answer, with correct units³"]
G --> H
Worked Examples
Example 1: Triangular Prism
A triangular prism has a cross-section that is a right-angled triangle with base 4 cm and perpendicular height 3 cm. The length of the prism is 10 cm. Find its volume.
- Step 1: Identify the cross-section. It's a triangle.
- Step 2: Calculate the area of the cross-section (the triangle).
Area of triangle = 12 × base × height = 12 × 4 cm × 3 cm = 6 cm². - Step 3: Identify the length of the prism. Length = 10 cm.
- Step 4: Apply the formula.
Volume = Area of cross-section × Length = 6 cm² × 10 cm = 60 cm³.
Example 2: Cylinder
A cylindrical tin of soup has a radius of 3.5 cm and a height of 11 cm. Calculate its volume. Give your answer to the nearest whole number.
- Step 1: It's a cylinder, so use V = πr²h.
- Step 2: Substitute the values. r = 3.5 cm, h = 11 cm.
V = π × (3.5)² × 11 - Step 3: Calculate carefully.
(3.5)² = 12.25
12.25 × 11 = 134.75
V = π × 134.75 ≈ 423.35... - Step 4: Round to the nearest whole number. Volume ≈ 423 cm³.
Dealing with Compound Shapes and Missing Lengths
Sometimes, questions combine shapes or hide information. The key is still to find the area of the cross-section first.
Example 3: L-Shaped Prism (Compound Cross-Section)
An L-shaped prism is made from two cuboids. The face (cross-section) is an 'L' shape: a 5 cm by 5 cm square with a 3 cm by 2 cm rectangle missing from one corner. The depth of the prism is 8 cm. Find the volume.
Step 1: Find the area of the L-shaped cross-section.
Area of full square = 5 cm × 5 cm = 25 cm².
Area of missing rectangle = 3 cm × 2 cm = 6 cm².
- Area of cross-section = 25 cm² - 6 cm² = 19 cm².
- Step 2: Identify the length/depth = 8 cm.
- Step 3: Volume = 19 cm² × 8 cm = 152 cm³.
Common Pitfalls and How to Avoid Them
Using the wrong "length": Ensure the length you multiply by is perpendicular to the cross-section. In a cylinder, it's the height, not a slant length.
Mixing units: Always convert all measurements to the same unit (e.g., all in cm or all in m) before calculating. If a prism's cross-section is in cm² and its length is in metres, you'll get the wrong answer.
Radius vs. Diameter: For cylinders, the formula uses radius². If you are given the diameter, you must halve it first. A very common error is using the diameter by mistake.
Units for Volume: Volume is measured in cubic units (e.g., cm³, m³). Always write your final answer with the correct cubic unit.
- Rounding too early: If a question asks for an answer to a certain degree of accuracy, only round your final answer. Keep the full value of π (or at least several decimal places) in your calculator during the calculation.
Practice for Your GCSE Exam
Try this typical GCSE-style question:
> A concrete beam is a hexagonal prism. The area of the hexagonal cross-section is 240 cm². The beam is 4.2 metres long. Calculate the volume of the beam in cm³.
Think: The cross-sectional area is given! But watch the units... the length is in metres, the area is in cm². We need consistent units.
Solution:
Length = 4.2 m = 4.2 × 100 = 420 cm.
Volume = Area × Length = 240 cm² × 420 cm = 100,800 cm³.
Summary and Key Takeaways
The volume of any prism is found by Area of cross-section × Length.
A cylinder is a prism with a circular cross-section: V = πr²h.
- Your step-by-step method is:
- 1) Identify shape/cross-section,
- 2) Calculate cross-section area,
- 3) Identify perpendicular length,
- 4) Multiply.
- Pay close attention to units and whether you have the radius or diameter.
- Mastering this topic is about applying a clear, consistent method. Break down the 3D shape, handle the 2D area calculation, and then bring it back to three dimensions. Good luck with your revision!