Volume of a Cuboid

\( V=\ell\,w\,h \)
Geometry GCSE
Question 7 of 20
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\( Find the volume of a cuboid with l=6 cm, w=4 cm, h=2 cm. \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( Use V=lwh. \)

Explanation

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Statement

The volume of a cuboid (a 3D box shape) is found by multiplying its three dimensions: length, width, and height.

\[ V = lwh \]

This formula works for any rectangular box where all faces are rectangles.

Why it’s true

  • Area of the base rectangle = length × width.
  • Volume = base area × height.
  • So volume = (l × w) × h = lwh.

Recipe (how to use it)

  1. Identify the three dimensions: length, width, height.
  2. Multiply them together.
  3. Include units (cm³, m³, etc.).

Spotting it

Look for 3D box shapes in questions with dimensions given.

Common pairings

  • Surface area problems (different formula).
  • Unit conversions (cm³ to m³, litres, etc.).

Mini examples

  1. Given: l=5 cm, w=3 cm, h=2 cm. Find: Volume. Answer: 30 cm³.
  2. Given: l=10 m, w=4 m, h=2 m. Find: Volume. Answer: 80 m³.

Pitfalls

  • Mixing up surface area and volume.
  • Forgetting units should be cubic.
  • Misreading one side as slanted when it’s actually vertical.

Exam strategy

  • Write the formula before substituting values.
  • Double-check all units are consistent.
  • If dimensions are given in mixed units, convert first.

Summary

The volume of a cuboid is simply the product of its length, width, and height. Always remember to cube the units when presenting your answer.

Worked examples

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  1. \( Find the volume of a cuboid with l=5 cm, w=3 cm, h=2 cm. \)
    1. \( V=lwh \)
    2. \( V=5×3×2=30 \)
    Answer: 30 cm³
  2. \( Find the volume of a cuboid with l=10 m, w=4 m, h=2 m. \)
    1. \( V=10×4×2=80 \)
    Answer: 80 m³
  3. \( Find the volume of a cuboid with l=7 cm, w=5 cm, h=3 cm. \)
    1. \( V=7×5×3=105 \)
    Answer: 105 cm³
  4. \( Find the volume of a cuboid with l=12 m, w=6 m, h=2 m. \)
    1. \( V=12×6×2=144 \)
    Answer: 144 m³
  5. \( Find the volume of a cuboid with l=8 cm, w=2 cm, h=10 cm. \)
    1. \( V=8×2×10=160 \)
    Answer: 160 cm³
  6. \( Find the volume of a cuboid with l=15 m, w=3 m, h=4 m. \)
    1. \( V=15×3×4=180 \)
    Answer: 180 m³
  7. \( Find the volume of a cuboid with l=2.5 m, w=1.2 m, h=0.5 m. \)
    1. \( V=2.5×1.2×0.5=1.5 \)
    Answer: 1.5 m³
  8. \( A cuboid has dimensions l=20 cm, w=15 cm, h=10 cm. Find the volume. \)
    1. \( V=20×15×10=3000 \)
    Answer: 3000 cm³
  9. \( Find the volume of a cuboid with l=100 cm, w=50 cm, h=20 cm. \)
    1. \( V=100×50×20=100000 \)
    Answer: 100000 cm³
  10. \( Find the volume of a cuboid with l=a, w=b, h=c. \)
    1. \( V=abc \)
    Answer: abc