# Circle

In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference. Usually however, the circumference means the length of the circle, and the interior of the circle is called a disk or disc.

## Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that

${\displaystyle \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.}$

If the circle is centered at the origin (0, 0), then this formula can be simplified to

${\displaystyle x^{2}+y^{2}=r^{2}.}$

The circle centered at the origin with radius 1 is called the unit circle.

Expressed in polar coordinates, (xy) can be written as

x = a + r cos(θ)
y = b + r sin(θ).

The slope a circle at a point (xy) can be expressed with the following formula:

${\displaystyle y'=-{\frac {x}{y}}.}$

In the complex plane, a circle with a center at c and radius r has the equation ${\displaystyle |z-c|^{2}=r^{2}}$. Since ${\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}$, the slightly generalized equation ${\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}$ for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words:

• Length of a circle's circumference = ${\displaystyle 2\pi \times r.}$
• Area of a circle = ${\displaystyle \pi \times r^{2}.}$

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

## Properties

### Chord properties

• Chords equidistant from the centre of a circle are equal.
• Equal chords are equidistant from the centre.
• A line from the centre, perpendicular to a chord, bisects the chord.
• The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
• The perpendicular bisector of a chord passes through the centre of a circle.

### Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

### Inscribed angle theorem

• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
• If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
• An inscribed angle subtended by a semicircle is a right angle.
• For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

### Secant, tangent, and chord properties

• The chord theorem states that if two chords, CD and EF, intersect at G, then ${\displaystyle CG\times DG=EG\times FG}$. (Chord Theorem)
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then ${\displaystyle DC^{2}=DG\times DE}$. (Tangent Secant Theorem)
• If two secants, DG and DE, also cut the circle at H and F respectively, then ${\displaystyle DH\times DG=DF\times DE}$. (Corollary of the Tangent Secant Theorem)