Circle

I

Introduction

Circle, in geometry, a two-dimensional curve such that each point on the curve is the same distance from a fixed point, called the center. The term circle may also be applied to the region enclosed by this curve.

II

Terms Used in Describing Circles

Any line that touches the circumference of a circle at one point, and one point only, is called a tangent. The line is said to be tangent to the circle at the point it touches. Any line that cuts through a circle, intersecting it at two points, is called a secant. The part of a secant within a circle is called a chord. If a chord passes through the center of a circle, it is called a diameter. The diameter is the longest possible chord for any given circle. Half a diameter, or the distance from the center of a circle to its edge, is a radius. The term radius can be used to indicate the distance, rather than an actual line, from the center to the circumference. Similarly the term diameter can be used to indicate the maximum width of a circle.

An arc of a circle is a portion lying between two points on the circle. A central angle is an angle with the vertex at the center of the circle and with sides forming radii of the circle. Concentric circles are circles that have the same center but different diameters.

III

Circumference and Area

The circumference—distance around the edge—of a circle is equal to a constant, pi (symbol p), times the circle’s diameter: C = pd. Since the diameter of a circle is equal to twice the circle’s radius, the circumference also equals two times pi times the radius: C = 2pr. Pi is one of the most important mathematical constants, and plays a role in many calculations and proofs in mathematics, physics, engineering, and other sciences. The first ten digits of pi are 3.141592654, although the approximations 3.14 or 3‡ are sufficiently accurate for many calculations.

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Of all two-dimensional figures having the same perimeter, the circle has the greatest area. The area of a circle is equal to pi multiplied by the square of the circle’s radius: A = pr².

IV

Circles as Conic Sections

The circle belongs to the class of curves known as conic sections because a circle can be described as the intersection of a right circular cone with a plane that is perpendicular to the axis of the cone. The equation that corresponds to this description for a circle with radius r and center (h, k) is: (x – h)² + (y - k)² = r². The equation (x – 2)² + (y - 1)² = 2², for example, represents a circle with a radius of 2 and center at (2, 1):

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