The edge of a cube measures r cm. The largest possible right circular cylinder is cut out of the cube. What do you think is the volume of the cylinder (in cm³)?

Step-by-step solution

Idea: The biggest cylinder stands on one face: its circular base touches all four sides of the square face, so its diameter equals the edge r, and its height is also r.

1. The base circle fits inside a square face of side r, so its diameter = r and radius = r2. The height of the cylinder = edge of the cube = r.1 mark
2. Volume = π × (radius)² × height = π × r²4 × r = πr³4 cm³.1 mark

Check: For r = 2: cylinder radius 1, height 2, volume 2π ≈ 6.28; formula π × 8 ÷ 4 = 2π ✓; less than the cube’s 8 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Taking the radius as r (the whole edge). The circle must fit inside the face, so the diameter is r.
• Writing πr³ ÷ 2 by squaring only the r and not the 2 in r2.

How this can come in the exam

Try one yourself

What fraction of the cube’s volume is wasted when the largest cylinder is cut from it?

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