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Cylinder volume and curved surface area

2026-09-22Spatial GeometryMs Tam

Separate volume from surface area so you do not mix πr²h with 2πrh.

Key formulas

V = πr²h
curved surface area = 2πrh
total surface area = 2πrh + 2πr²

Two different questions

Volume fills the interior. Curved surface area wraps the side. Total surface area adds the two circular ends. Read the wording carefully before choosing a formula.

Worked example

A closed cylinder has radius 4 cm and height 10 cm. Volume = π × 16 × 10 = 160π cm³. Curved surface area = 2π × 4 × 10 = 80π cm². Total surface area = 80π + 2π × 16 = 80π + 32π = 112π cm².

Open cylinder warning

If the cylinder is open at one end, total surface area becomes 2πrh + πr². Always ask how many circular faces are included.

Worked example

Worked example

Problem. A cylinder has radius 4 cm and height 10 cm. Find its volume and curved surface area in terms of π.

Solution. V = πr²h = π × 16 × 10 = 160π cm³. Curved surface = 2πrh = 2π × 4 × 10 = 80π cm².

Quick numbers

r = 3, h = 8 → V = 72π. Curved area = 48π.

Review quiz

Cylinder r=3, h=7. Volume in terms of π is:

Practice questions

  1. Find the volume and curved surface area of a cylinder with r = 5 cm and h = 12 cm.
  2. A cylinder is open at the top. Find its total surface area when r = 6 and h = 9.

Study notes

Why keep answers in terms of π?

The question often asks for an exact value, and it prevents calculator rounding.

How do I remember 2πrh?

Unroll the side into a rectangle of width 2πr and height h.

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