GCSE — mathsHigher2 min read

Volume and Surface Area

Calculate volumes and surface areas of 3D shapes for GCSE Maths: prisms, cylinders, cones, spheres, and pyramids.

Tutor AI Team
Tutor AI Team

Volume measures the space inside a 3D shape. Surface area measures the total area of all its faces. Both are key GCSE topics.

Formulas

Prisms

V=cross-section area×lengthV = \text{cross-section area} \times \text{length}

Cylinder

V=πr2hV = \pi r^2 h, SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi rh

Cone

V=13πr2hV = \frac{1}{3}\pi r^2 h, SA=πrl+πr2SA = \pi r l + \pi r^2 (where ll = slant height)

Sphere

V=43πr3V = \frac{4}{3}\pi r^3, SA=4πr2SA = 4\pi r^2

Pyramid

V=13×base area×hV = \frac{1}{3} \times \text{base area} \times h

Worked Example: Cylinder

Problem

r=5r = 5, h=10h = 10. V=π(25)(10)=250π≈785.4V = \pi(25)(10) = 250\pi \approx 785.4 cm³.

Solution

Worked Example: Sphere

Problem

r=6r = 6. V=43π(216)=288π≈904.8V = \frac{4}{3}\pi(216) = 288\pi \approx 904.8 cm³.

Solution

Worked Example: Cone

Problem

r=4r = 4, h=9h = 9. V=13π(16)(9)=48π≈150.8V = \frac{1}{3}\pi(16)(9) = 48\pi \approx 150.8 cm³.

Solution

Worked Example: Triangular Prism

Problem

Cross-section: triangle base 6, height 4. Length 10. V=12(6)(4)×10=120V = \frac{1}{2}(6)(4) \times 10 = 120 cm³.

Solution

Practice Problems

    1. Volume of cylinder: r=3r = 3, h=7h = 7.
    1. Surface area of sphere: r=5r = 5.
    1. Volume of cone: r=6r = 6, h=8h = 8.
    1. A hemisphere has radius 10. Find its volume.

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Key Takeaways

  • ✓

    Prism: cross-section × length.

  • ✓

    Cone/Pyramid: 13\frac{1}{3} of corresponding prism/cylinder.

  • ✓

    Sphere: V=43πr3V = \frac{4}{3}\pi r^3, SA=4πr2SA = 4\pi r^2.

  • ✓

    Hemisphere: half the sphere, plus circular base for surface area.

Tags:
#gcse#maths#geometry#volume#surface-area#3d

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