How to Find the Central Angle of a Circle

A central angle in a circle is a fundamental geometric concept that quantitatively describes the angle formed at the center of a circle by two radii extending to distinct points on its circumference. The relationship among the central angle, arc length, and radius is established through rigorous geometric reasoning.

Mathematical Definition of the Central Angle of a Circle

Consider a circle with center $O$ and radius $r$. Let points $A$ and $B$ lie on the circumference such that the radii $\overline{OA}$ and $\overline{OB}$ together form an angle $\theta$ at $O$. This angle $\theta$, measured in radians or degrees, is termed the central angle subtended by arc $AB$ at the center of the circle.

Expression for Central Angle in Radians

Let the length of the arc $AB$ be denoted as $s$. The measure of the central angle $\theta$ in radians is given by the ratio of the arc length to the radius, expressed as

$\theta = \dfrac{s}{r}$.

This formula directly relates the arc length $s$ to the radius $r$ and the central angle $\theta$ when $\theta$ is measured in radians.

Expression for Central Angle in Degrees

To obtain the central angle in degrees, observe that the circumference of a circle is $C = 2\pi r$ and a complete revolution about the center corresponds to $360^{\circ}$. Hence, the proportionality becomes:

$\dfrac{\theta}{360^{\circ}} = \dfrac{s}{2\pi r}$

Multiplying both sides by $360^{\circ}$ gives

$\theta = \dfrac{s \times 360^{\circ}}{2\pi r}$

where

$\theta$ = central angle (in degrees)
$s$ = arc length
$r$ = radius of the circle.

Derivation of the Central Angle Formulas

Let the length of arc $s$ subtend a central angle $\theta$ at $O$. The complete circumference corresponds to a central angle of $2\pi$ radians or $360^{\circ}$ degrees.

In radians, the circular arc length formula provides $s = r\theta$. To isolate $\theta$, divide both sides by $r$:

$s = r\theta$

$\dfrac{s}{r} = \theta$

Result: $\theta = \dfrac{s}{r}$ radians

For degrees, start with the proportion between the arc length and circumference and the corresponding angles:

$\dfrac{s}{2\pi r} = \dfrac{\theta}{360^{\circ}}$

Multiply both sides by $360^{\circ}$:

$\theta = \dfrac{s \times 360^{\circ}}{2\pi r}$

Result: $\theta = \dfrac{s \times 360^{\circ}}{2\pi r}$ degrees

Arc Length Formula can be used in conjunction with the above for related calculations.

Theorem Relating Central and Inscribed Angles

For any two distinct points $A$ and $B$ on the circumference, the central angle $\theta$ subtended at $O$ by arc $AB$ is always exactly double the inscribed angle $\alpha$ subtended at any other point $P$ lying on the remaining arc $AB$. That is,

$\theta = 2\alpha$

This result is fundamental in circle geometry and is a direct consequence of the properties of arcs and angles in circles.

Computation of Radius or Arc Length from Central Angle

The relationships can be rearranged based on known and unknown quantities. To solve for $r$ given $s$ and $\theta$ (in radians):

$r = \dfrac{s}{\theta}$

Alternatively, if $\theta$ is in degrees, express it as

$r = \dfrac{s \times 360^{\circ}}{2\pi \theta}$

Similarly, to compute the arc length from the central angle (in radians):

$s = r\theta$

Worked Examples on the Central Angle of a Circle

Example 1: Given arc length $s = 30$ units and radius $r = 15$ units, determine the central angle in degrees.

Substitute these into the degree formula:

$\theta = \dfrac{30 \times 360^{\circ}}{2\pi \times 15}$

$2\pi \times 15 = 30\pi$

$\theta = \dfrac{30 \times 360^{\circ}}{30\pi}$

$30$ cancels in numerator and denominator:

$\theta = \dfrac{360^{\circ}}{\pi}$

Numerically, $\pi \approx 3.1416$

$\theta \approx \dfrac{360^{\circ}}{3.1416} \approx 114.59^{\circ}$

Result: $\theta \approx 114.59^{\circ}$

Area of a Circle Formula is frequently cross-utilized with central angle problems.

Example 2: The central angle formed is $\theta = 90^\circ$ and the arc length is $s = 12$ cm. Find the radius $r$.

Using the formula:

$\theta = \dfrac{s \times 360^{\circ}}{2\pi r}$

$90^{\circ} = \dfrac{12 \times 360^{\circ}}{2\pi r}$

To isolate $r$, first multiply both sides by $2\pi r$:

$90^{\circ} \times 2\pi r = 12 \times 360^{\circ}$

$180\pi r = 4320$

Now, divide both sides by $180\pi$:

$r = \dfrac{4320}{180\pi}$

Calculate this value:

$4320 \div 180 = 24$

$r = \dfrac{24}{\pi} \approx \dfrac{24}{3.1416} \approx 7.64$ cm

Result: $r \approx 7.64$ cm

Interpretation of Central Angles and Arcs

An arc less than $180^\circ$ is known as a minor arc, while an arc greater than $180^\circ$ is termed a major arc. If the arc measures exactly $180^\circ$, it is a semicircle. The value of the central angle unambiguously determines the type of arc.

Summary of Key Formulas

In radians: $\theta = \dfrac{s}{r}$

In degrees: $\theta = \dfrac{s \times 360^{\circ}}{2\pi r}$

For more advanced geometric constructions involving sectors or permutations, reference to Circular Permutation is suggested.