Volume

Calculating the space inside 3D shapes

GCSE Geometry And Measures
Volume – cover
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Volume

Measuring the space inside 3D shapes

Introduction

Volume tells us how much space a 3D solid takes up. It’s measured in cubic units like cm³, m³, or litres. At GCSE, you’ll work with the volumes of cuboids, prisms, cylinders, cones, spheres, and compound solids. Volume links maths with real life: filling a tank, packaging boxes, or working out how much concrete is needed.

Why it matters: Volume is about capacity. If you can calculate it, you can solve problems about containers, construction, storage, and design.

Key Vocabulary

  • Volume — the space a 3D solid occupies, measured in cubic units.
  • Prism — a solid with the same cross-section throughout (e.g. cuboid, cylinder, triangular prism).
  • Cuboid — a box-shaped solid with length, width, and height.
  • Cylinder — a prism with a circular cross-section.
  • Sphere — perfectly round 3D shape, like a ball.
  • Cone — solid with a circular base tapering to a point (apex).
  • Pyramid — solid with a polygon base and triangular faces meeting at an apex.
  • Units — cm³, m³ for volume; 1 cm³ = 1 ml, 1000 cm³ = 1 litre.

Core Ideas

  • Volume of a cuboid = length × width × height.
  • Volume of a prism = area of cross-section × length.
  • Volume of a cylinder = πr²h.
  • Volume of a cone = \(\tfrac{1}{3}πr²h\).
  • Volume of a sphere = \(\tfrac{4}{3}πr³\).
  • Volume of a pyramid = \(\tfrac{1}{3}\) × base area × height.
  • Compound solids = split into known shapes, calculate separately, then add or subtract.

Step-by-Step Method

  1. Identify the shape (cuboid, cylinder, cone, sphere, prism, pyramid).
  2. Recall the correct formula.
  3. Substitute measurements (make sure all in same units).
  4. Square or cube values where needed.
  5. Multiply carefully, use π ≈ 3.1416 or calculator π button.
  6. Round final answer to given accuracy and include units (cm³, m³, litres).

Worked Examples — Foundation

Example F1 — Cuboid

[diagram: cuboid with length 5 cm, width 4 cm, height 3 cm]

Volume = \(5×4×3=60\text{ cm}³\).

Example F2 — Triangular Prism

[diagram: triangular cross-section with base 6 cm, height 4 cm, prism length 10 cm]

Area of cross-section = ½ × 6 × 4 = 12 cm². Volume = 12 × 10 = 120 cm³.

Example F3 — Cylinder

[diagram: cylinder with radius 7 cm, height 10 cm]

Volume = πr²h = π×7²×10 = π×49×10 = 490π ≈ 1538.6 cm³.

Worked Examples — Higher

Example H1 — Cone

[diagram: cone radius 6 cm, height 9 cm]

Volume = ⅓πr²h = ⅓π×6²×9 = ⅓π×36×9 = 108π ≈ 339.3 cm³.

Example H2 — Sphere

[diagram: sphere radius 5 cm]

Volume = 4/3 πr³ = 4/3 π×125 = 500/3 π ≈ 523.6 cm³.

Example H3 — Compound Shape

[diagram: cylinder with cone on top, radius 4 cm, cylinder height 10 cm, cone height 6 cm]

Cylinder = πr²h = π×16×10 = 160π. Cone = ⅓πr²h = ⅓π×16×6 = 32π. Total = 192π ≈ 603.2 cm³.

Common Mistakes & Fixes

  • Mixing units (e.g. cm and m). Always convert first.
  • Forgetting to cube for volume. Don’t use area formulas.
  • Using diameter instead of radius. Remember r = d/2.
  • Missing the ⅓ in cones and pyramids. Write it down clearly.
  • Leaving out π or approximating too early. Keep π until the final step for accuracy.

Practice Questions — Foundation

  1. A cuboid has dimensions 8 cm × 5 cm × 2 cm. Find its volume.
  2. A triangular prism has cross-section base 10 cm, height 6 cm, and prism length 12 cm. Find the volume.
  3. A cylinder has radius 3 cm and height 15 cm. Work out the volume (1 dp).
  4. A box is 20 cm long, 15 cm wide, 10 cm high. How many litres is its volume? (1 litre = 1000 cm³).
  5. A triangular prism has cross-section area 25 cm² and length 9 cm. Find its volume.

Practice Questions — Higher

  1. A cone has radius 5 cm, height 12 cm. Find its volume (1 dp).
  2. A sphere radius 7 cm. Find its exact volume in terms of π.
  3. A pyramid with square base 9 cm × 9 cm and height 15 cm. Find its volume.
  4. A water tank is a cylinder radius 1.2 m, height 3 m. Work out volume in litres. (1 m³ = 1000 litres).
  5. A solid is made of a cylinder radius 4 cm, height 8 cm, with a hemisphere on top. Find its volume.

Challenge Questions

  1. A cone of radius 4 cm and height 9 cm is melted and recast into a sphere. Find the sphere’s radius (2 dp).
  2. A cylinder has diameter 10 cm and height 12 cm. A hole of radius 2 cm runs through it (like a pipe). Find the remaining volume (1 dp).
  3. A cuboid 40 cm × 25 cm × 30 cm is filled with sand. How many 5 litre buckets can be filled? (1 litre = 1000 cm³).

Quick Revision Sheet

  • Cuboid: \(V=lwh\).
  • Prism: \(V=\text{area of cross-section}×\text{length}\).
  • Cylinder: \(V=πr^2h\).
  • Cone: \(V=\tfrac{1}{3}πr^2h\).
  • Sphere: \(V=\tfrac{4}{3}πr^3\).
  • Pyramid: \(V=\tfrac{1}{3}\text{base area}×h\).
  • Conversions: 1 cm³=1 ml; 1000 cm³=1 litre; 1 m³=1000 litres.

Conclusion & Next Steps

Volume connects geometry to real-world measurement. Once you’re fluent with these formulas, you’ll be ready for surface area, density, mass calculations, and exam questions mixing different solids.

Exam Tips

  • Underline the shape type before starting.
  • Always use radius, not diameter.
  • Show the formula before substituting values.
  • Keep π until the last step for accuracy.
  • Attach correct units (cm³, m³, litres).

Answer Key (Outline)

Foundation

  1. 80 cm³
  2. 360 cm³
  3. 424.1 cm³
  4. 3000 cm³ = 3 litres
  5. 225 cm³

Higher

  1. 314.2 cm³
  2. \(\tfrac{1372}{3}π\) cm³
  3. 405 cm³
  4. 13,570 litres
  5. \(402π\) ≈ 1263.0 cm³
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