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Circles

Circles are fundamental geometric shapes defined as the set of all points in a plane that are equidistant from a fixed center point.

Basic Definitions

  • Circle: The set of all points in a plane at a fixed distance (radius) from a center point
    • Alternative definition: Three points that do not lie on a common straight line (i.e., they form a triangle) uniquely define a plane circle (the circumcircle of the triangle)
  • Center: The fixed point from which all points on the circle are equidistant
  • Radius: The distance from the center to any point on the circle, denoted by $r$
  • Diameter: A line segment passing through the center with endpoints on the circle, denoted by $d = 2r$
  • Chord: A line segment with both endpoints on the circle
  • Arc: A portion of the circle's circumference between two points
  • Circumference: The total distance around the circle, denoted by $C$
  • Degree: $1^\circ$ is 1/360 of a complete rotation around a circle

Fundamental Measurements

Circumference

The circumference of a circle is the distance around it, given by: $C = 2\pi r = \pi d$ where:

  • circumference: $C$
  • radius: $r$
  • diameter: $d$
  • pi / $\pi$ ≈ 3.14159...

Key Properties:

  • The ratio of circumference to diameter is always $\pi$, regardless of the circle's size
  • This relationship is the defining property of $\pi$
  • For a circle with radius $r = 1$, the circumference is $2\pi$

Area

The area enclosed by a circle is: $A = \pi r^2$ where:

  • area: $A$
  • radius: $r$

Derivation:

The area formula can be derived by dividing the circle into infinitely many thin triangular sectors. Each sector has:

  • Base = infinitesimal arc length
  • Height = radius $r$

When summed, the total base length equals the circumference $2\pi r$, giving:

\[\begin{aligned} A &= \sum \frac{1}{2} \times \text{base} \times \text{height} \\ &= \frac{1}{2} \times 2\pi r \times r \\ &= \pi r^2 \end{aligned}\]

Alternative formulas:

  • In terms of diameter: $A = \frac{\pi d^2}{4}$
  • In terms of circumference: $A = \frac{C^2}{4\pi}$

Parts of a Circle

Chords and Arcs

  • Chord: Any line segment connecting two points on the circle
    • The diameter is the longest chord
    • A chord divides the circle into two arcs
  • Arc: A portion of the circumference
    • Minor arc: The shorter arc between two points (less than $180^\circ$)
    • Major arc: The longer arc between two points (greater than $180^\circ$)
    • Semicircle: An arc that is exactly half the circle ($180^\circ$)

Angle Measurement: Radians and Degrees

Definitions

  • Radian: A unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius

    • The radian is the natural unit for measuring angles in mathematics
    • A full circle contains $2\pi$ radians
    • One radian ≈ 57.2958°
    • One degree = $\frac{1}{360}*2\pi = \frac{\pi}{180}$ radians

  • Conversion between radians and degrees:

    • To convert from degrees to radians: $\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$
    • To convert from radians to degrees: $\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}$

Common Angles

DegreesRadians
$90^\circ$$\frac{\pi}{2}$
$180^\circ$$\pi$
$270^\circ$$\frac{3\pi}{2}$
$360^\circ$$2\pi$

Arc Length

  • Arc Length: The length of an arc subtended by a central angle $\theta$ in a circle of radius $r$ is given by:
    • For angle $\theta$ (in radians): $s = r\theta$
    • For angle $\theta$ (in degrees): $s = \frac{\pi r \theta}{180}$

Sectors and Segments

  • Sector: The region bounded by two radii and the arc between them (like a "slice of pie")

    • Area of sector with central angle $\theta$ (in radians): $A_{\text{sector}} = \frac{1}{2}r^2\theta$

    • Area of sector with central angle $\theta$ (in degrees): $A_{\text{sector}} = \frac{\pi r^2 \theta}{360}$

  • Segment: The region bounded by a chord and the arc it subtends

    • Area of segment with central angle $\theta$ (in radians): $A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2}r^2(\theta - \sin\theta)$
    • Area of segment with central angle $\theta$ (in degrees): $A_{\text{segment}} = \frac{\pi r^2 \theta}{360} - \frac{1}{2}r^2\sin\left(\frac{\pi \theta}{180}\right)$

Circle Equations

Standard Form (Cartesian)

A circle with center $(h, k)$ and radius $r$ has equation: $(x - h)^2 + (y - k)^2 = r^2$

Special case: Circle centered at origin $(0, 0)$: $x^2 + y^2 = r^2$

Parametric Form

A circle with center $(h, k)$ and radius $r$ can be expressed parametrically:

\[\begin{aligned} x &= h + r\cos\theta \\ y &= k + r\sin\theta \end{aligned}\]

where $\theta$ ranges from $0$ to $2\pi$ (or $0^\circ$ to $360^\circ$)

General Form

The general quadratic equation: $x^2 + y^2 + Dx + Ey + F = 0$

represents a circle when the coefficients of $x^2$ and $y^2$ are equal. Converting to standard form:

  • Center: $\left(-\frac{D}{2}, -\frac{E}{2}\right)$
  • Radius: $r = \sqrt{\frac{D^2 + E^2}{4} - F}$

Angles in Circles

Central Angle

  • Central Angle: An angle whose vertex is at the center of the circle
  • The measure of a central angle equals the measure of its intercepted arc
  • A central angle of $\theta$ radians intercepts an arc of length $s = r\theta$

Inscribed Angle

  • Inscribed Angle: An angle formed by two chords with vertex on the circle
  • Inscribed Angle Theorem: An inscribed angle is half the measure of its intercepted arc $\theta_{\text{inscribed}} = \frac{1}{2}\theta_{\text{central}}$

Angle Relationships

  • Angles subtended by the same arc: All inscribed angles subtending the same arc are equal
  • Thales' Theorem: An angle inscribed in a semicircle is always a right angle ($90^\circ$)

Common Circle Theorems

Chord Properties

  1. Perpendicular from center: A perpendicular from the center to a chord bisects the chord
  2. Equal chords: Chords equidistant from the center are equal in length
  3. Chord length formula: For a chord at distance $d$ from center in a circle of radius $r$: $\text{chord length} = 2\sqrt{r^2 - d^2}$

Tangent Properties

  • Tangent: A line that touches the circle at exactly one point
  • Tangent perpendicularity: A tangent is perpendicular to the radius at the point of tangency
  • Tangent length: For an external point $P$ at distance $d$ from center, tangent length is: $\text{tangent length} = \sqrt{d^2 - r^2}$
  • Two tangents from external point: Tangent segments from an external point to a circle are equal in length

Applications

Geometry and Trigonometry

  • Unit circle: A circle with radius 1, fundamental to trigonometry
  • Coordinate geometry: Circles are conic sections (intersection of cone and plane)
  • Angle measurement: Radians are defined using the unit circle

Real-World Applications

  • Engineering: Wheels, gears, pulleys
  • Architecture: Arches, domes, circular structures
  • Physics: Circular motion, orbits, waves
  • Navigation: Distance calculations on Earth's surface
  • Design: Circular patterns, logos, decorative elements

Further Information

More information about circles and their properties can be found in: