Circle

Definition

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The constant distance from the center to any point on the circle is called the radius.

Mathematically, a circle with center at point (h, k) and radius r is defined as:

(x - h)² + (y - k)² = r²

Key Properties

Radius (r): Distance from center to any point on the circle.
Diameter (d): Distance across the circle through the center; d = 2r.
Circumference (C): Distance around the circle; C = 2πr = πd.
Area (A): Region enclosed by the circle; A = πr².
Arc: A connected portion of the circumference.
Chord: A line segment connecting two points on the circle.
Secant: A line intersecting the circle at two points.
Tangent: A line touching the circle at exactly one point.

Formulas

Basic Formulas

Diameter: d = 2r

Circumference: C = 2πr = πd

Area: A = πr²

Arc Length

For a central angle θ (in radians):

Arc Length: s = rθ

For a central angle θ (in degrees):

Arc Length: s = (θ/360°) × 2πr

Sector Area

For a central angle θ (in radians):

Sector Area: A = (1/2)r²θ

Segment Area

Segment Area = Sector Area - Triangle Area

Important Theorems

1. Inscribed Angle Theorem

An angle inscribed in a circle is half the measure of its intercepted arc.

Inscribed Angle = (1/2) × Central Angle = (1/2) × Arc Measure

2. Thales' Theorem

If A, B, and C are points on a circle where AC is a diameter, then angle ABC is a right angle (90°).

3. Tangent-Radius Theorem

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

4. Chord Properties

The perpendicular from the center to a chord bisects the chord.
Equal chords are equidistant from the center.
The line joining the center to the midpoint of a chord is perpendicular to the chord.

Circle Equations

Standard Form

(x - h)² + (y - k)² = r²

Center: (h, k), Radius: r

General Form

x² + y² + Dx + Ey + F = 0

Center: (-D/2, -E/2), Radius: √(D²/4 + E²/4 - F)

Examples

Example 1: Find the circumference and area of a circle with radius 5 cm.

Solution:
Circumference: C = 2πr = 2π(5) = 10π ≈ 31.42 cm
Area: A = πr² = π(5)² = 25π ≈ 78.54 cm²

Example 2: Find the equation of a circle with center at (3, -2) and radius 4.

Solution:
Using standard form: (x - h)² + (y - k)² = r²
(x - 3)² + (y - (-2))² = 4²
(x - 3)² + (y + 2)² = 16

Example 3: Find the arc length for a central angle of 60° in a circle with radius 12 cm.

Solution:
s = (θ/360°) × 2πr
s = (60°/360°) × 2π(12)
s = (1/6) × 24π = 4π ≈ 12.57 cm

Applications

Engineering: Design of wheels, gears, pipes, and circular structures.
Architecture: Domes, arches, and circular buildings.
Physics: Circular motion, orbital mechanics, and wave propagation.
Computer Graphics: Rendering circles and curves, collision detection.
Navigation: GPS systems use circular geometry for positioning.
Art and Design: Aesthetically pleasing proportions and compositions.