Definition:Circle
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 a part of its circumference between two given points.
Hence for two given points $A$ and $B$ on the circumference of a circle, there are two such arcs so defined.
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 $\phi = \dfrac \pi 2$, 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 $R$ whose center is at the origin:
- in Cartesian coordinates is $x^2 + y^2 = R^2$
- in polar coordinates is $\map r \theta = R$
- parametrically can be expressed as $x = R \cos t, y = R \sin t$.
- Results about circles can be found here.
Linguistic Note
The adjectival form of the word circle is circular.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): circle
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circle
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): circle
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): circle
- For a video presentation of the contents of this page, visit the Khan Academy.