Central Angle on Arc

Definition of Central Angle

A central angle is formed by two radii and has its vertex at the center of the circle. A slice of pizza gives the same shape: its pointed tip lies at the center, and its two straight edges follow radii.

In a circle with center $O$, let $A$ and $B$ lie on the circumference. Then $\angle AOB$ is a central angle, and it subtends the arc $AB$ between those two points.

Relationship Between Central Angle and Arc

A central angle and its intercepted arc have the same degree measure. A $60^\circ$ central angle therefore intercepts a $60^\circ$ arc.

If $\angle AOB = 60^\circ$, then arc $AB$ also measures $60^\circ$. Both numbers are degree measures. The arc length uses a length unit such as centimeters.

Types of Arcs Based on Central Angle

Based on the measure of their central angles, arcs fall into three types. The central angle decides which fraction of the circumference the arc covers.

Minor Arc

A minor arc is subtended by a central angle less than $180^\circ$. It is shorter than half the circumference. At $90^\circ$ it covers a quarter of the circumference, because $90/360$ is exactly one quarter.

Semicircle Arc

A semicircular arc is subtended by a central angle of exactly $180^\circ$. Its endpoints lie on a diameter, so its length is half the circle's circumference.

Major Arc

A major arc is subtended by a reflex central angle greater than $180^\circ$. It is longer than half the circumference. Its degree measure is the remainder of the full turn after subtracting the corresponding minor arc measure:

Arc Length Calculation

Arc length can be calculated using the ratio between the central angle and the full angle of the circle. The basic formula for calculating arc length is:

The variables in this formula are:

• $s$ = arc length
• $\theta$ = central angle measure in degrees
• $r$ = radius of the circle

A radian measures arc length divided by radius, so an angle in radians gives the formula directly:

Here $\theta$ must be measured in radians.

Calculating Arc Length from a Central Angle

Consider a circle with radius $r = 6 \text{ cm}$ and central angle $\theta = 120^\circ$.

Substitute the radius and central angle into the arc-length formula:

The arc corresponding to the central angle $120^\circ$ is therefore $4\pi \text{ cm}$ long, or approximately $12.57 \text{ cm}$.

Examples of Central Angles

On a gear or disk, parts separated by equal central angles are spaced evenly. To calculate the distance between those parts along the edge, we also need the radius. On an analog clock, the hands sweep central angles as time passes. Arc geometry can also model the shape of an arch or dome, but structural design requires additional information.

If Earth is modeled as a sphere, a route along a great circle has length $s=r\theta$ for a central angle $\theta$ in radians. This spherical model is sufficient for a mathematical example, but precise navigation requires an Earth model and route data with suitable accuracy.

Exercises

Each problem relates a central angle to the arc it cuts, so choose between degrees and radians and keep one unit throughout the calculation.

1. A circle has a radius of $8 \text{ cm}$. If the central angle facing an arc is $45^\circ$, calculate the length of that arc.

2. Given that the arc length of a circle is $10\pi \text{ cm}$ and its radius is $15 \text{ cm}$. Determine the measure of the central angle facing that arc.

3. In a circle with center $O$, there is a central angle $\angle AOB = 72^\circ$. If the circle's radius is $r = 5 \text{ cm}$, determine the length of arc $AB$ and express the result in terms of $\pi$.

4. A bicycle wheel with radius $30 \text{ cm}$ rotates about a fixed axle through $150^\circ$. How far does a point on its rim move along its circular path?

Worked Solutions

1. Solution:

   Given: $r = 8 \text{ cm}$ and $\theta = 45^\circ$

   Find: Arc length $s$

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

   $$s = \frac{45^\circ}{360^\circ} \times 2\pi \times 8$$

   Step 3: Simplify the fraction

   $$s = \frac{1}{8} \times 16\pi$$

   Step 4: Calculate the final result

   $$s = 2\pi \text{ cm}$$

   Therefore, the arc length is $2\pi \text{ cm}$ or approximately $6.28 \text{ cm}$.

2. Solution:

   Given: $s = 10\pi \text{ cm}$ and $r = 15 \text{ cm}$

   Find: Central angle measure $\theta$

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

   $$10\pi = \frac{\theta}{360^\circ} \times 2\pi \times 15$$

   Step 3: Simplify the equation

   $$10\pi = \frac{\theta \times 30\pi}{360^\circ}$$

   Step 4: Isolate $\theta$

   $$\theta = \frac{10\pi \times 360^\circ}{30\pi}$$

   Step 5: Calculate the final result

   $$\theta = \frac{3600^\circ}{30} = 120^\circ$$

   Therefore, the central angle measure facing that arc is $120^\circ$.

3. Solution:

   Given: $\angle AOB = 72^\circ$ and $r = 5 \text{ cm}$

   Find: Length of arc $AB$

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

   $$s = \frac{72^\circ}{360^\circ} \times 2\pi \times 5$$

   Step 3: Simplify the fraction

   $$s = \frac{72^\circ}{360^\circ} \times 10\pi = \frac{1}{5} \times 10\pi$$

   Step 4: Calculate the final result

   $$s = 2\pi \text{ cm}$$

   The length of arc $AB$ is $2\pi \text{ cm}$.

4. Solution:

   Given: $r = 30 \text{ cm}$ and $\theta = 150^\circ$

   Find: Distance traveled (arc length)

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

   $$s = \frac{150^\circ}{360^\circ} \times 2\pi \times 30$$

   Step 3: Simplify the fraction

   $$s = \frac{150^\circ}{360^\circ} \times 60\pi = \frac{5}{12} \times 60\pi$$

   Step 4: Calculate the final result

   $$s = 25\pi \text{ cm}$$

   The point moves $25\pi \text{ cm}$ along its circular path, or approximately $78.54 \text{ cm}$.