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Learn · Mathematics · Grade 10 · Geometry

Prisms, cylinders, cones and spheres

Surface area and volume of the standard solids, nets, and how scaling a solid scales its area and volume.

High school Lesson 22 of 31 Interactive figure Practice

First, the rules

Base area times height — then a third, then a sphere

Every "straight" solid (prism, cylinder) is base area × height. Every "pointed" solid (pyramid, cone) is one third of that. The sphere has formulas of its own.

★ Prism and cylinder
Example 1
V = (base area) · h
box: abc, cylinder: πr²h
★ Pyramid and cone
Example 2
V = 13 · (base area) · h
cone: 13πr²h
★ Sphere
Example 3
V = 43πr³, A = 4πr²
Surface areas
Example 2
box: 2(ab + bc + ac)
cylinder: 2πr² + 2πrh (two discs + the unrolled rectangle)
cone: πr² + πrl (l = slant height = √(r² + h²))
★ Scaling
Example 3
Scale every length by k: areas × k², volumes × k³. Double the radius of a ball: 4 × the surface, 8 × the volume.
Writing answers
Leave π in the answer (12π) unless a decimal is asked for; the practice box accepts both.

See it

Turn the dimensions

Pick a solid and stretch it. Volume and surface update live, in exact π form where it exists, and the sketch keeps its dimensions labelled.

Step by step

Worked examples

1

A can

cylinder

A cylindrical can has radius 4 cm and height 10 cm. Find its volume and the area of metal needed to make it (closed top and bottom).

Ask yourself: Volume is base × height. The surface is two discs plus a rectangle whose width is the circumference.

Show the solution
  1. Volume: π · 4² · 10 = 160π ≈ 502.7 cm³.
  2. Surface: 2π · 4² + 2π · 4 · 10 = 32π + 80π = 112π ≈ 351.9 cm².

Answer V = 160π cm³, A = 112π cm²

2

A cone from a right triangle

cone · slant height

A cone has radius 6 and height 8. Find its volume and its total surface area.

Ask yourself: The slant height closes a 6-8-10 right triangle.

Show the solution
  1. Volume: 13 · π · 6² · 8 = 96π.
  2. Slant height l = √6² + 8² = 10.
  3. Surface: base π · 6² = 36π plus lateral π · 6 · 10 = 60π: 96π in total.

Answer V = 96π, A = 96π

3

Scaling a sphere

sphere · k³

A sphere has radius 3. (a) Find its volume and surface area. (b) A second sphere has twice the radius. How many times larger are its surface area and its volume?

Ask yourself: Compute once, then think in ratios: lengths × 2 means areas × 2² and volumes × 2³.

Show the solution
  1. (a) V = 43π · 27 = 36π, A = 4π · 9 = 36π.
  2. (b) Surface × 4 (= 144π), volume × 8 (= 288π). Check directly: r = 6 gives 4π · 36 = 144π and (4/3)π · 216 = 288π. ✓

Answer 36π, 36π; ×4 and ×8

Practice: solids

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Volumes and surface areas of boxes, cylinders, cones and spheres — type 50π or 157.08, both are accepted. Your best score is kept in this browser only.

About this lesson

Prisms, cylinders, pyramids, cones and spheres: volume and surface area formulas, nets, and how scaling a solid by k multiplies areas by k² and volumes by k³. Answers can be typed with π (like 50π) or as decimals. Worked examples and practice.

Part of Learn mathematics · Grade 10 · Geometry. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.