Area & Volume: Prisms, Cylinders, Cones, Spheres & Frustums

GCSE Maths · Geometry

Area & Volume

You need to know formulas for a range of 2D areas and 3D volumes. Some are given on the exam formula sheet; others you must memorise.

2D Areas to Memorise

Shape	Formula
Rectangle	length × width
Triangle	½ × base × height
Parallelogram	base × perpendicular height
Trapezium	½ × (a + b) × h (where a, b are parallel sides)
Circle	π × r²

Prisms

A prism has a uniform cross-section. Its volume is:

Volume of a prism = area of cross-section × length

Worked Example: A triangular prism has a cross-section that is a right-angled triangle with base 6 cm and height 4 cm. The prism is 10 cm long.

Cross-section area = ½ × 6 × 4 = 12 cm²
Volume = 12 × 10 = 120 cm³

Cylinders

A cylinder is a prism with a circular cross-section.

Volume = π × r² × h
Curved surface area = 2 × π × r × h
Total surface area = 2πrh + 2πr² (curved surface + two circular ends)

Worked Example: A cylinder has radius 5 cm and height 12 cm.

Volume = π × 25 × 12 = 300π ≈ 942.5 cm³
Total SA = 2π(5)(12) + 2π(25) = 120π + 50π = 170π ≈ 534.1 cm²

Cones (Given on Formula Sheet)

Volume = ⅓ × π × r² × h
Curved surface area = π × r × l (where l is the slant height)

Worked Example: A cone has radius 3 cm and height 4 cm. Find the volume and curved surface area.

Volume = ⅓ × π × 9 × 4 = 12π ≈ 37.7 cm³
Slant height: l = √(3² + 4²) = √25 = 5 cm
Curved SA = π × 3 × 5 = 15π ≈ 47.1 cm²

Spheres (Given on Formula Sheet)

Volume = 4/3 × π × r³
Surface area = 4 × π × r²

Worked Example: A sphere has radius 6 cm.

Volume = 4/3 × π × 216 = 288π ≈ 904.8 cm³
Surface area = 4 × π × 36 = 144π ≈ 452.4 cm²

Hemispheres

A hemisphere is half a sphere.

Volume = ⅔ × π × r³
Curved surface area = 2πr²
Total surface area = 2πr² + πr² = 3πr² (curved + flat circle)

Frustums (Higher)

A frustum is a cone with the top cut off. Find the volume by subtracting the small cone from the large cone.

Worked Example: A frustum is made by cutting a cone of height 12 cm and radius 6 cm. The cut is made 8 cm from the base, leaving a top radius of 2 cm and a height of 4 cm for the removed cone.

Large cone volume = ⅓π(6²)(12) = 144π
Small cone volume = ⅓π(2²)(4) = 16π/3
Frustum volume = 144π − 16π/3 = 432π/3 − 16π/3 = 416π/3 ≈ 435.6 cm³

Surface Area of Complex Shapes

Break the shape into parts, find each area, and add. Remember:

When shapes are joined, remove the areas where they connect
A hemisphere on top of a cylinder: total SA = curved cylinder + base circle + curved hemisphere (no circle between them)

Worked Example: A solid is made from a cylinder (r = 4, h = 10) with a hemisphere on top. Find total SA.

Cylinder curved SA = 2π(4)(10) = 80π
Cylinder base = π(16) = 16π
Hemisphere curved SA = 2π(16) = 32π
Total = 80π + 16π + 32π = 128π ≈ 402.1 cm²

Exam Tips

Formulas for cones, spheres and pyramids are on the formula sheet — know where to find them
Always use the perpendicular height in volume formulas, not the slant height
For frustums, use similar triangles to find the dimensions of the small cone if not given
Leave answers in terms of π when the question says "give an exact answer"
When finding density or mass, calculate volume first, then use mass = density × volume