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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/central-angle-on-arc
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/central-angle-on-arc/en.mdx

Learn central angles formed by radii, their relationship with arcs, and how to classify and calculate angle-arc measures.

---

## 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 $$B$$.

Visible text: In a circle with center , if we have points and on the circumference of the circle, then is a central angle. This angle faces arc which is the part of the circle's circumference between points and .

Component: LineEquation
Props:
- title: Central Angle Visualization $$\angle AOB$$
  Visible text: Central Angle Visualization
- description: Central angle formed by two radii $$OA$$ and{" "}
$$OB$$ with vertex at the center of circle{" "}
$$O$$.
  Visible text: Central angle formed by two radii and{" "}
 with vertex at the center of circle{" "}
.
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3 * Math.cos(Math.PI / 6), y: 3 * Math.sin(Math.PI / 6), z: 0 },
],
color: getColor("ORANGE"),
labels: [{ text: "OA", at: 0.5, offset: [0.3, 0.3, 0] }],
showPoints: true,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3 * Math.cos(5 * Math.PI / 6), y: 3 * Math.sin(5 * Math.PI / 6), z: 0 },
],
color: getColor("ORANGE"),
labels: [{ text: "OB", at: 0.5, offset: [-0.3, 0.3, 0] }],
showPoints: true,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const angle = (Math.PI / 6) + (i * (2 * Math.PI / 3) / 20);
return {
x: 1.5 * Math.cos(angle),
y: 1.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
lineWidth: 2,
showPoints: false,
labels: [{ text: "θ = 120°", at: 10, offset: [0.5, 0.5, 0] }],
},
{
points: Array.from({ length: 61 }, (_, i) => {
const angle = (Math.PI / 6) + (i * (2 * Math.PI / 3) / 60);
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("VIOLET"),
lineWidth: 3,
showPoints: false,
labels: [{ text: "Arc AB", at: 30, offset: [0.5, 0.5, 0] }],
},
{
points: Array.from({ length: 101 }, (_, i) => {
const angle = i * (2 * Math.PI / 100);
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0, ... [truncated; 1284 chars]
- cameraPosition: [0, 0, 10]
- showZAxis: false

## 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.

```math
\text{Central angle measure} = \text{Arc measure faced}
```

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

Visible text: If the central angle , then arc also measures . 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 $$180^\circ$$. This is an arc shorter than half a circle. Like a piece of cake smaller than half the whole cake.

Visible text: A minor arc is an arc faced by a central angle less than . 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 $$180^\circ$$. This central angle is formed by the diameter of the circle, so the arc is half the circumference of the circle.

Visible text: A semicircle arc is faced by a central angle of exactly . 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 $$180^\circ$$. This is an arc longer than half a circle. To calculate the measure of a major arc, we use:

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

Component: ContentStack
Children:

```math
\text{Major arc} = 360^\circ - \text{Minor arc}
```

Component: LineEquation
Props:
- title: Comparison of Arc Types
- description: Visualization of minor arc, semicircle, and major arc.
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const angle = i * (Math.PI / 3) / 20;
return {
x: 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
lineWidth: 2,
showPoints: false,
labels: [{ text: "θ = 60°", at: 10, offset: [1, 0.3, 0] }],
},
{
points: Array.from({ length: 31 }, (_, i) => {
const angle = i * (Math.PI / 3) / 30;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("LIME"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Minor Arc (60°)", at: 15, offset: [1, 0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(Math.PI / 3), y: 4 * Math.sin(Math.PI / 3), z: 0 },
],
color: getColor("ORANGE"),
lineWidth: 2,
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
lineWidth: 2,
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
lineWidth: 2,
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const angle = Math.PI + (i * Math.PI / 20);
return {
x: 2.5 * Math.cos(angle),
y: 2.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
lineWidth: 2,
showPoint ... [truncated; 1804 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

## 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:

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

Where:

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

Visible text: - = arc length
- = central angle measure in degrees
- = radius of the circle

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

```math
s = \theta \times r
```

Where $$\theta$$ is in radians.

Visible text: Where 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 $$\theta = 120^\circ$$.

Visible text: Let's apply this concept in a concrete example. Suppose we have a circle with radius and central angle .

First step, we calculate the arc length:

Component: MathContainer
Children:

```math
s = \frac{120^\circ}{360^\circ} \times 2\pi \times 6
```

```math
s = \frac{1}{3} \times 12\pi
```

```math
s = 4\pi \text{ cm}
```

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

Visible text: So the arc length faced by the central angle is or approximately .

Component: LineEquation
Props:
- title: Central Angle Example $$120^\circ$$
  Visible text: Central Angle Example
- description: Circle with radius $$6 \text{ cm}$$ and central angle $$120^\circ$$.
  Visible text: Circle with radius and central angle .
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 6, y: 0, z: 0 },
],
color: getColor("PURPLE"),
labels: [{ text: "r = 6 cm", at: 0.5, offset: [0, -0.5, 0] }],
showPoints: true,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 6 * Math.cos(2 * Math.PI / 3), y: 6 * Math.sin(2 * Math.PI / 3), z: 0 },
],
color: getColor("PURPLE"),
labels: [{ text: "r = 6 cm", at: 0.5, offset: [-0.5, 0.3, 0] }],
showPoints: true,
},
{
points: Array.from({ length: 25 }, (_, i) => {
const angle = i * (2 * Math.PI / 3) / 24;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("ORANGE"),
lineWidth: 2,
showPoints: false,
labels: [{ text: "θ = 120°", at: 12, offset: [0.8, 0.8, 0] }],
},
{
points: Array.from({ length: 41 }, (_, i) => {
const angle = i * (2 * Math.PI / 3) / 40;
return {
x: 6 * Math.cos(angle),
y: 6 * Math.sin(angle),
z: 0,
};
}),
color: getColor("ROSE"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "s = 4π cm", at: 20, offset: [-1, -1, 0] }],
},
{
points: Array.from({ length: 101 }, (_, i) => {
const angle = i * (2 * Math.PI / 100);
return {
x: 6 * Math.cos(angle),
y: 6 * Math.sin(angle),
z: 0,
};
}),
color: getColor("INDIGO"),
lineWidth: 1,
showPoints: false,
smooth: tr ... [truncated; 1208 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

## 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 $$8 \text{ cm}$$. 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 $$5 \text{ cm}$$, determine the length of arc $$AB$$ and express the result in terms of $$\pi$$.

4. A bicycle wheel has a radius of $$30 \text{ cm}$$. If the wheel rotates forming a central angle of $$150^\circ$$, what distance is traveled by a point on the wheel's edge?

Visible text: 1. A circle has a radius of . If the central angle facing an arc is , calculate the length of that arc.

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

3. In a circle with center , there is a central angle . If the circle's radius is , determine the length of arc and express the result in terms of .

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

### Answer Key

1. **Solution:**

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

   Find: Arc length $$s$$

   **Step** $$1$$: Use the arc length formula

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

   **Step** $$2$$: Substitute the known values

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

   **Step** $$3$$: Simplify the fraction

   
   
   ```math
   s = \frac{1}{8} \times 16\pi
   ```

   **Step** $$4$$: Calculate the final result

   
   
   ```math
   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

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

   **Step** $$2$$: Substitute the known values

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

   **Step** $$3$$: Simplify the equation

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

   **Step** $$4$$: Isolate θ

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

   **Step** $$5$$: Calculate the final result

   
   
   ```math
   \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

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

   **Step** $$2$$: Substitute the known values

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

   **Step** $$3$$: Simplify the fraction

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

   **Step** $$4$$: Calculate the final result

   
   
   ```math
   s = 2\pi \text{ cm}
   ```

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

4. **Solution:**

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

   Find: Distance traveled (arc length)

   **Step** $$1$$: Use the arc length formula

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

   **Step** $$2$$: Substitute the known values

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

   **Step** $$3$$: Simplify the fraction

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

   **Step** $$4$$: Calculate the final result

   
   
   ```math
   s = 25\pi \text{ cm}
   ```

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

Visible text: 1. **Solution:**

 Given: and 

 Find: Arc length 

 **Step** : Use the arc length formula

 
 

 **Step** : Substitute the known values

 
 

 **Step** : Simplify the fraction

 
 

 **Step** : Calculate the final result

 
 

 Therefore, the arc length is or approximately .

2. **Solution:**

 Given: and 

 Find: Central angle measure 

 **Step** : Use the arc length formula

 
 

 **Step** : Substitute the known values

 
 

 **Step** : Simplify the equation

 
 

 **Step** : Isolate θ

 
 

 **Step** : Calculate the final result

 
 

 Therefore, the central angle measure facing that arc is .

3. **Solution:**

 Given: and 

 Find: Length of arc 

 **Step** : Use the arc length formula

 
 

 **Step** : Substitute the known values

 
 

 **Step** : Simplify the fraction

 
 

 **Step** : Calculate the final result

 
 

 Therefore, the length of arc is .

4. **Solution:**

 Given: and 

 Find: Distance traveled (arc length)

 **Step** : Use the arc length formula

 
 

 **Step** : Substitute the known values

 
 

 **Step** : Simplify the fraction

 
 

 **Step** : Calculate the final result

 
 

 Therefore, the distance traveled by a point on the wheel's edge is or approximately .