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Circles (Mathematics Glossary)

What Is a Circle?

A circle is a shape containing a set of points that are all the same distance from its center.



Dictionary Definition

The Oxford English Dictionary defines a circle as "a plane figure bounded by a single curved line, called the circumference, which is everywhere equally distant from a point within, called the centre. But often applied to the circumference alone, without the included space."

Parts of a Circle

  • The circle technically refers to the boundary of the shape (that is the curved blue line).


  • The center is a point inside the circle. All points on the circle are the same distance from the center.

Radius, Diameter and Circumference

The most important lengths in a circle are the radius, the diameter and the circumference.
  • The radius is the line segment from the center of the circle to any point on the circle.

    The radius also refers to the length of this line. The radius is usually denoted by the symbol r.


  • The diameter is the line segment that contains the center of the circle and has its endpoints on the circle.

    The diameter also refers to the length of this line. The diameter is usually denoted by the symbol d.



    The diameter is twice the length of the radius. d = 2r

  • The circumference is the distance around the circle.

    The circumference is usually denoted by the symbol C.



    The circumference is π × the length of the diameter. C = πd.

Other Parts of a Circle

  • A chord is a line whose endpoints lie on the circle.



    Note: The diameter is the chord that contains the center.

  • An arc is a portion of the circle.


  • A tangent is a line that touches the circle at one point.


  • A sector is a region bounded by two radii and the arc lying between the radii.

    A sector looks like a slice of cake.


  • A segment is a region, not containing the center, bounded by a chord and an arc lying between the chord's endpoints.

Properties of Circles

Circumference of a Circle

The circumference of a circle is found using the formula:



In the formula, d is the diameter of the circle. The image above shows what we mean by the diameter.

Read more about how to find the circumference of a circle

Area of a Circle

The area of a circle is found using the formula:



In the formula, r is the radius of the circle. The image above shows what we mean by the radius.

Read more about how to find the area of a circle

Equation of a Circle

The equation of a circle is:



In the equation, (x, y) are the Cartesian coordinates of the points on the circle. (a, b) are the Cartesian coordinates of the center of the circle and r is the radius of the circle.

The image below shows what we mean by a point on the circle, the center and the radius of the circle:



Read more about the equation of a circle

Circle Geometry

There are several theorems about circle geometry.



Read more about circle theorems

Curriculum
Geometry Lessons
Interactive Test
  show
 


Here's a second test on the circle.
Here's a third test on the circle.
Note

What's in a Name?

Circle comes from the Greek word 'kirkos' or 'kuklos', meaning 'hoop' or 'ring'.

This became the Latin word 'circus', also meaning 'ring'.

The word 'circulus' refered to a small ring. This became the Old French word 'cercle', which meant a ring (for the finger).

Drawing a Circle

The circle has the special property that all points on it are the same distance from the circle's centre.

This makes it easy to draw a circle - just keep the pencil the same length from the center.

Push a pin into a piece of paper. Attach a pencil to the pin by a length of string. Keeping the string taut, draw around in a circle.



A compass is often used to draw a circle. It keeps the pencil a fixed distance from a point. This distance can be adjusted to draw circles with a different radius.

What Is a Circle Exactly?

We all know what a circle looks like, but its technical definition is more precise.

A circle refers to the boundary of the shape, not the space within the boundary. However, in everyday language we call the whole shape a circle.

The boundary and the space within it should technically be refered to as a 'disk'.

Circles as Conic Sections

A circle is found by slicing through a cone parallel to the base:



Other shapes, such as ellipses, parabolas and hyperbolas can also be found by slicing through a cone.

Euclid and Circles

Euclid defines circles in his book, Elements.



Definition 15: "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 equal one another."

Definition 16: "And the point is called the center of the circle."