Volume of Prisms and Cylinders
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is a Prism?
A prism is a 3D shape with two identical, parallel end faces (the cross-section), connected by flat side faces. The cross-section stays exactly the same all the way through the shape.
- Cuboid: a prism with a rectangular cross-section.
- Triangular prism: a prism with a triangular cross-section.
- Cylinder: a prism with a circular cross-section.
Deriving the Volume Formula
Because a prism's cross-section repeats identically along its whole length, its volume is simply the cross-sectional area multiplied by the length: Volume = cross-sectional area × length. This one idea underlies every prism - only the shape of the cross-section changes.
Volume = cross-sectional area × length, for any prism
Volume of a Cuboid
A cuboid's cross-section is a rectangle, so its volume is V = length × width × height.
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Question: Find the volume of a cuboid 8 cm × 5 cm × 4 cm.
- Step 1: V = 8 × 5 × 4
- Answer: 160 cm³
Volume of a Triangular Prism
A triangular prism's cross-section is a triangle, so first find the triangle's area (½ × base × height), then multiply by the prism's length.
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Question: A triangular prism has a triangular cross-section with base 6 cm and height 4 cm, and the prism is 10 cm long. Find its volume.
- Step 1: Cross-sectional area = ½ × 6 × 4 = 12 cm²
- Step 2: Volume = 12 × 10
- Answer: 120 cm³
Volume of a Cylinder
A cylinder's cross-section is a circle, so its volume is V = πr² × h.
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Question: Find the volume of a cylinder with radius 5 cm and height 12 cm (use π ≈ 3.14).
- Step 1: Cross-sectional area = πr² = 3.14 × 5² = 78.5 cm²
- Step 2: Volume = 78.5 × 12
- Answer: 942 cm³
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Question: Find the volume of a cylindrical tank with diameter 8 m and height 3 m (use π ≈ 3.14).
- Step 1: Halve the diameter to find the radius: r = 4 m
- Step 2: Cross-sectional area = πr² = 3.14 × 4² = 50.24 m²
- Step 3: Volume = 50.24 × 3
- Answer: 150.72 m³
Finding a Missing Dimension from Volume
If the volume is known, rearrange the volume formula to find a missing length, width, height, or radius.
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Question: A cuboid has volume 240 cm³, length 8 cm, and width 5 cm. Find its height.
- Step 1: 240 = 8 × 5 × h, so 240 = 40 × h
- Step 2: Divide both sides by 40: h = 240 ÷ 40
- Answer: h = 6 cm
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Question: A cylinder has volume 471 cm³ and radius 5 cm. Find its height (use π ≈ 3.14).
- Step 1: Cross-sectional area = πr² = 3.14 × 25 = 78.5 cm²
- Step 2: 471 = 78.5 × h, so h = 471 ÷ 78.5
- Answer: h = 6 cm
Real-World Applications
Prism and cylinder volumes are used in many practical contexts:
- Water tanks and pipes: cylindrical volume calculations.
- Packaging boxes: cuboid volume for shipping and storage.
- Tents and roof trusses: triangular prism shapes.
- Concrete beams and columns: volume for material estimates.
- Food and drink containers: capacity of cylindrical cans and cartons.
Exam Tips
Common Mistakes
MistakeForgetting the ½ in the triangle's cross-sectional area
Fixcross-sectional area of a triangular prism is always ½ × base × height, before multiplying by the length
MistakeConfusing the triangle's own height with how long the whole prism is
Fixidentify the cross-section's dimensions separately from the prism's length
MistakeUsing the diameter instead of the radius in the cylinder volume formula
Fixalways halve the diameter to get the radius before using V = πr²h
MistakeRounding π too early in a multi-step calculation
Fixuse π ≈ 3.14 throughout and round only the final answer
MistakeMixing units, e.g. combining cm and m without converting
Fixconvert all dimensions to the same unit before calculating volume
For Exams
- Remember the core idea: Volume = cross-sectional area × length, for any prism.
- Learn each cross-section formula: rectangle, triangle, circle.
- For cylinders: find the radius first if you're given the diameter.
- Show two steps: find the cross-sectional area, then multiply by length.
- Check your units: volume is always measured in cubed units (cm³, m³).
Interactive revision notes, videos and practice questions load below.