Surface Area and Volume of a Cylinder

A cylinder is one of the most common 3D shapes -- think of cans, pipes, and columns. In this article you will learn every formula you need and work through three complete examples.

---

Table of contents

• What is a cylinder?
• Key dimensions
• Net of a cylinder
• Base area
• Lateral (side) area
• Total surface area
• Volume
• Worked example 1 -- Total surface area
• Worked example 2 -- Volume
• Worked example 3 -- Finding the radius
• Summary of formulas
• Practice exercises

---

What is a cylinder?

A cylinder is a 3D solid with:

• Two parallel circular bases of equal size
• A curved lateral surface connecting the bases

The bases are always congruent circles, and the lateral surface wraps around them like a label on a tin can.

---

Key dimensions

Every cylinder is described by two measurements:

• Radius $r$ -- the radius of each circular base
• Height $h$ -- the perpendicular distance between the two bases

      ┌─────────────┐
     ╱       r       ╲
    ╱    ●───────      ╲      ● = centre
   │                    │
   │         h          │
   │                    │
    ╲                  ╱
     ╲               ╱
      └─────────────┘

The diameter is $d=2r$. If a problem gives you the diameter, remember to halve it to get $r$.

---

Net of a cylinder

If you "unroll" a cylinder and lay it flat, you get its net:

• Two circles (top and bottom bases)
• One rectangle whose width equals the circumference of the base ($2\pi r$) and whose height equals $h$

    ┌──── 2πr ────┐
    │              │
  h │  (rectangle) │
    │              │
    └──────────────┘

       ○  (circle, radius r)  -- top base
       ○  (circle, radius r)  -- bottom base

Understanding the net makes it easy to see where every part of the surface area formula comes from.

---

Base area

Each base is a circle with radius $r$:

Since a cylinder has two bases, the combined base area is:

---

Lateral (side) area

The lateral surface, when unrolled, is a rectangle of width $2\pi r$ and height $h$:

This is the area of the "label" that wraps around the cylinder.

---

Total surface area

The total surface area is the sum of both bases and the lateral surface:

This can be factored neatly:

Use $S=2\pi r(r+h)$ as your go-to formula for the total surface area of a cylinder.

---

Volume

The volume of a cylinder equals the base area multiplied by the height:

This makes intuitive sense: you are "stacking" circles of area $\pi r^2$ up to a height of $h$.

---

Worked example 1 -- Total surface area

Problem: A cylinder has radius $r=5\text{cm}$ and height $h=12\text{cm}$. Find its total surface area. Solution:

---

Worked example 2 -- Volume

Problem: A cylindrical water tank has diameter $d=1.2\text{m}$ and height $h=2\text{m}$. How many litres of water can it hold? Solution:

First, find the radius: $r=\frac{d}{2}=\frac{1.2}{2}=0.6\text{m}$.

Convert to litres ($1{\text{m}}^3=1000\text{l}$):

The tank can hold approximately 2 262 litres of water.

---

Worked example 3 -- Finding the radius

Problem: A cylinder has volume $V=1000{\text{cm}}^3$ and height $h=8\text{cm}$. Find the radius. Solution:

Start from the volume formula and solve for $r$:

---

Summary of formulas

---

Related articles

• Surface Area and Volume -- Introduction
• Pyramid -- Surface Area and Volume
• Cone -- Surface Area and Volume
• Sphere -- Surface Area and Volume
• Formula Reference Sheet

---

Practice exercises

Put your knowledge to the test:

• Cylinder -- Surface Area Exercises
• Cylinder -- Volume Exercises