# Definition:Circle

## Contents

## Definition

In the words of Euclid:

*A***circle**is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

(*The Elements*: Book $\text{I}$: Definition $15$)

### Center

In the words of Euclid:

*And the point is called the***center of the circle**.

(*The Elements*: Book $\text{I}$: Definition $16$)

In the above diagram, the **center** is the point $A$.

### Circumference

The **circumference** of a circle is the line that forms its boundary.

### Diameter

In the words of Euclid:

*A***diameter of the circle**is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.

(*The Elements*: Book $\text{I}$: Definition $17$)

In the above diagram, the line $CD$ is a **diameter**.

### Radius

A **radius** of a circle is a straight line segment whose endpoints are the center and the circumference of the circle.

In the above diagram, the line $AB$ is a **radius**.

### Arc

An **arc** of a circle is any part of its circumference.

## Semicircle

In the words of Euclid:

*A***semicircle**is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

(*The Elements*: Book $\text{I}$: Definition $18$)

## Chord

A **chord** of a circle is a straight line segment whose endpoints are on the circumference of the circle.

In the diagram above, the lines $CD$ and $EF$ are both **chords**.

## Equality

In the words of Euclid:

**Equal circles**are those the diameters of which are equal, or the radii of which are equal.

(*The Elements*: Book $\text{III}$: Definition $1$)

### Equidistant from Center

In the words of Euclid:

*In a circle straight lines are said to be***equally distant from the center**when the perpendiculars drawn to them from the center are equal.

(*The Elements*: Book $\text{III}$: Definition $4$)

*And that straight line is said to be***at a greater distance**on which the greater perpendicular falls.

(*The Elements*: Book $\text{III}$: Definition $5$)

## Intersection with Cone

Let $C$ be a double napped right circular cone whose base is $B$.

Let $\theta$ be half the opening angle of $C$.

That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.

Let a plane $D$ intersect $C$.

Let $\phi$ be the inclination of $D$ to the axis of $C$.

Let $K$ be the set of points which forms the intersection of $C$ with $D$.

Then $K$ is a **conic section**, whose nature depends on $\phi$.

Let $\phi = \dfrac \pi 2 - \theta$, thereby making $D$ perpendicular to the axis of $C$.

Then $D$ and $B$ are parallel, and so $K$ is a circle.

## Also see

- Area of Circle: the area of a
**circle**is $\pi r^2$, where $r$ is its radius. - Perimeter of Circle: the perimeter of a
**circle**is $2 \pi r$, where $r$ is its radius.

- Equation of Circle: the equation of a circle of radius $R$:

- in Cartesian coordinates is $x^2 + y^2 = R^2$
- in polar coordinates is $r \left({\theta}\right) = R$
- parametrically can be expressed as $x = R \cos t, y = R \sin t$.

- Results about
**circles**can be found here.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies

- For a video presentation of the contents of this page, visit the Khan Academy.