# Definition:Cylinder

## Contents

## Definition

A **cylinder** is a solid made by rotating a rectangle along one of its sides.

In the words of Euclid:

*When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a***cylinder**.

(*The Elements*: Book $\text{XI}$: Definition $21$)

In the above diagram, the rectangle $ADHG$ has been rotated around the side $GH$ to produce the **cylinder** $ACBEFD$.

### Axis of Cylinder

In the words of Euclid:

*The***axis of the cylinder**is the straight line which remains fixed and about which the parallelogram is turned.

(*The Elements*: Book $\text{XI}$: Definition $22$)

In the above diagram, the **axis** of the cylinder $ACBEFD$ is the straight line $GH$.

### Base of Cylinder

In the words of Euclid:

*And the***bases**are the circles described by the two sides opposite to one another which are carried round.

(*The Elements*: Book $\text{XI}$: Definition $23$)

In the above diagram, the **bases** of the cylinder $ACBEDF$ are the faces $ABC$ and $DEF$.

### Height of Cylinder

The **height** of a cylinder is the length of a line segment drawn perpendicular to the base and its opposite plane.

In the above diagram, $h$ is the **height** of the cylinder $ACBDFE$.

## Similar Cylinders

Let $h_1$ and $h_2$ be the heights of two cylinders.

Let $d_1$ and $d_2$ be the diameters of the bases of the two cylinders.

Then the two cylinders are **similar** if and only if:

- $\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$

In the words of Euclid:

**Similar cones and cylinders**are those in which the axes and the diameters of the bases are proportional.