Central Angle on Arc

Definition of Central Angle

A central angle is an angle formed by two radii of a circle with its vertex located at the center of the circle. Imagine it like a slice of pizza, where the pointed tip of the pizza is exactly at the center of the circle. Both sides of this central angle are radii that connect the center of the circle to the edge of the circle.

In a circle with center O, if we have points A and B on the circumference of the circle, then $\angle AOB$ is a central angle. This angle faces arc AB which is the part of the circle's circumference between points A and B.

Relationship Between Central Angle and Arc

The fundamental relationship between a central angle and the arc it faces is very simple yet important. The measure of a central angle in degrees equals the measure of the arc it faces in degrees. This is like a direct relationship between a bottle opener and the bottle cap being opened.

If the central angle $\angle AOB = 60°$, then arc AB also measures $60°$. This concept applies because an arc is defined based on the central angle that faces it.

Types of Arcs Based on Central Angle

Based on the measure of their central angles, arcs can be classified into three types:

Minor Arc

A minor arc is an arc faced by a central angle less than $180°$. This is an arc shorter than half a circle. Like a piece of cake smaller than half the whole cake.

Semicircle Arc

A semicircle arc is faced by a central angle of exactly $180°$. This central angle is formed by the diameter of the circle, so the arc is half the circumference of the circle.

Major Arc

A major arc is an arc faced by a central angle greater than $180°$. This is an arc longer than half a circle. To calculate the measure of a major arc, we use:

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:

Where:

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

If the central angle is expressed in radians, the formula becomes simpler:

Where $\theta$ is in radians.

Example Application

Let's apply this concept in a concrete example. Suppose we have a circle with radius $r = 6 \text{ cm}$ and central angle $\theta = 120°$.

First step, we calculate the arc length:

So the arc length faced by the central angle $120°$ is $4\pi \text{ cm}$ or approximately $12.57 \text{ cm}$.

Applications in Daily Life

The concept of central angles on arcs has many practical applications. In gear design, central angles determine the distance between teeth. On analog clocks, clock hands move forming central angles that indicate time. Architects use this concept to design bridge arches or building domes.

In navigation, aircraft pilots use the concept of circular arcs to calculate travel distance when flying along curved paths on Earth's surface. The larger the central angle traversed, the farther the distance traveled.

Exercises

1. A circle has a radius of 8 cm. If the central angle facing an arc is 45°, calculate the length of that arc.

2. Given that the arc length of a circle is 10π cm and its radius is 15 cm. Determine the measure of the central angle facing that arc.

3. In a circle with center O, there is a central angle $\angle AOB = 72°$. If the circle's radius is 5 cm, determine the length of arc AB and express the result in terms of π.

4. A bicycle wheel has a radius of 30 cm. If the wheel rotates forming a central angle of 150°, what distance is traveled by a point on the wheel's edge?

Answer Key

1. Solution:

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

   Find: Arc length $s$

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

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

   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 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

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

   Step 2: Substitute the known values

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

   Step 3: Simplify the equation

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

   Step 4: Isolate θ

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

   Step 5: Calculate the final result

   $$\theta = \frac{3600°}{30} = 120°$$

   Therefore, the central angle measure facing that arc is 120°.

3. Solution:

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

   Find: Length of arc AB

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

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

   Step 3: Simplify the fraction

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

   Step 4: Calculate the final result

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

   Therefore, the length of arc AB is $2\pi \text{ cm}$.

4. Solution:

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

   Find: Distance traveled (arc length)

   Step 1: Use the arc length formula

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

   Step 2: Substitute the known values

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

   Step 3: Simplify the fraction

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

   Step 4: Calculate the final result

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

   Therefore, the distance traveled by a point on the wheel's edge is $25\pi \text{ cm}$ or approximately 78.54 cm.