Definition:Circle

Definition
As defined by 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."


 * Circle.png

Center
As defined by Euclid:


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

(Note: UK English spells this centre.)

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

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

It is also often taken to refer to the length of this line.

Diameter
As defined by 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 circle."

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

Semicircle
As defined by 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.

Radius
A radius (plural radii, pronounced ray-dee-eye) 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.

Sector
A sector of a circle is the area bounded by two radii and an arc.

In the above diagram, $ABC$ is a sector.


 * Sector.png

In fact there are two sectors, together making up the whole of the circle.

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

In the diagram at the top of the page, the lines $CD$ and $EF$ are both chords.

Note that under this definition, the diameter itself is in fact a chord.

Equality
As defined by Euclid:


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

Equally Distant from the Center
As defined by 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."

Area
It can be shown that the area of a circle is $\pi r^2$, where $r$ is the radius.

Equation
It can be shown that the equation of a circle in Cartesian coordinates is $x^2 + y^2 = R^2$, in polar coordinates is $r \left({\theta}\right) = R$, and parametrically by $x = R \cos t, y = R \sin t$.