Circles — Mathematics Knowledge System

Overview

A circle is the set of all points in a plane that are at a fixed distance — the radius — from a fixed point called the center. The distance across the circle through the center is the diameter, exactly twice the radius. A circle encloses a two-dimensional region called a disc. Circles are governed by the constant π (pi), the ratio of a circle's circumference to its diameter, and they connect directly to angle measure, since a full rotation around a circle's center is 360° (2π radians).

Intuition

Tie one end of a string to a peg, pull the string taut, and trace the other end all the way around: every point you trace is exactly the string's length from the peg. That's a circle — the purest shape defined entirely by 'equal distance from one point.' Because every point on it is equidistant from the center, a circle has perfect rotational symmetry: it looks identical after any rotation about its center, which is why wheels, gears, and orbits are circular.

Interactive Graph

Formal Definition

Definition

In the coordinate plane, a circle with center (h, k) and radius r is the set of points (x, y) satisfying:

(x-h)^2 + (y-k)^2 = r^2

Follows directly from the distance formula / Pythagorean theorem

Standard equation of a circle

C = 2\pi r = \pi d

Circumference

A = \pi r^2

Area enclosed by a circle

s = r\theta

Arc length for central angle θ in radians

Notation

Notation	Meaning
$r$	Radius: distance from center to any point on the circle
$d = 2r$	Diameter: distance across the circle through the center
$\pi$	Pi, the ratio of circumference to diameter (≈ 3.14159...)
$C$	Circumference: the total distance around the circle

Properties

Chord

\text{A line segment with both endpoints on the circle; the diameter is the longest possible chord.}

Tangent line

\text{A line touching the circle at exactly one point, always perpendicular to the radius drawn to that point.}

Inscribed angle theorem

\text{An inscribed angle is half the central angle that subtends the same arc: } \theta_{\text{inscribed}} = \tfrac{1}{2}\theta_{\text{central}}

Angle in a semicircle

\text{An inscribed angle subtending a diameter is always } 90^\circ.

Example: Special case of the inscribed angle theorem with a 180° central angle.

Applications

Uniform circular motion (orbits, rotating machinery) is described directly using radius, angular velocity, and arc length relationships.

Worked Examples

Apply the circumference formula.
$C = 2\pi(5) = 10\pi \approx 31.4$
Apply the area formula.
$A = \pi(5)^2 = 25\pi \approx 78.5$

Answer: C = 10π ≈ 31.4, A = 25π ≈ 78.5

Practice Problems

Difficulty 2/10

A circle has circumference 20π. Find its radius and area.

Difficulty 3/10

A central angle of 60° subtends an arc on a circle of radius 9. Find the arc length.

Common Mistakes

Common Mistake

Confusing radius and diameter when plugging into A = πr².

Always convert diameter to radius first (r = d/2) before squaring — using the diameter directly gives an area four times too large.

Common Mistake

Using degrees directly in the arc-length formula s = rθ.

The formula s = rθ requires θ in radians. Convert degrees to radians first: θ_rad = θ_deg · π/180.

Summary

A circle is the set of points at fixed distance r (the radius) from a center point.
Circumference C = 2πr = πd; area A = πr².
Standard equation: (x-h)² + (y-k)² = r², derived from the distance formula.
An inscribed angle is always half the central angle subtending the same arc; an angle inscribed in a semicircle is always 90°.
Arc length s = rθ requires the central angle θ measured in radians.

References

WebsiteWikipedia — Circle