Circle

In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. 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. More usually, the circumference means the length of the circle, and the interior of the circle is called a disc.

In an x-y coordinate system, the circle with centre (x₀,y₀) and radius r is the set of all points (x,y) such that

(x - x₀)² + (y - y₀)² = r².

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

x² + y² = r².

A circle is a kind of conic section, with eccentricity zero. All circles are similar, so the ratio between the circumference and radius and that between the area and radius square are both constants. These are 2π and π, respectively, and these are the best known definitions of that constant. In other words:

• Length of a circle's circumference = 2 × π × radius
• Area of a circle = π × (radius)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, 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 two and dividing by 2, we get the area πr².

A line cutting a circle in two places is called a secant, and a line touching the circle in one place is called a tangent. The tangent lines are necessarily perpendicular to the radii, segments connecting the centre to a point on the circle, whose length matches the definition given above. The segment of a secant bound by the circle is called a chord, and the longest chord is that which passes through the centre, called the diameter and divided into two radii.

A segment of a circle bound by two radii is called an arc, and the ratio between the length of an arc and the radius defines the angle between the two radii in radians.

In affine geometry all circles and ellipses become (affinely) isomorphic, and in projective geometry the other conic sections join them. In topology all simple closed curves are homeomorphic to circles, and the word circle is often applied to them as a result. The 3-dimensional analog of the circle is the sphere.

Every triangle gives rise to several circles: its circumcircle containing all three vertices, its incircle lying inside the circle and touching all three sides, the three excircles lying outside the circle and touching one side and the extensions of the other two, and its nine point circle which contains various important points of the triangle.