# Nakafa Learning Content

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

Learn circle fundamentals, arc types, and sector calculations. Use circle elements to solve arc length problems.

---

## Definition of Circle

A circle is the set of all points on a plane that have the same distance to a fixed point. The fixed point is called the **center of the circle**, while the same distance is called the **radius**.

Mathematically, a circle with center $$O(a,b)$$ and radius $$r$$ can be expressed with the equation:

Visible text: Mathematically, a circle with center and radius can be expressed with the equation:

```math
(x-a)^2 + (y-b)^2 = r^2
```

### Elements of Circle

How is the visualization of the circle equation?

Component: LineEquation
Props:
- title: Elements of Circle
- description: Visualization of center, radius, diameter, and chord of circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "Circle", at: 0, offset: [1.5, -1, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [0, -0.5, 0] },
{ text: "r", at: 0, offset: [1.5, 0.3, 0] },
],
},
{
points: [
{ x: -3, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("CYAN"),
labels: [{ text: "diameter", at: 0, offset: [0, -0.5, 0] }],
},
{
points: [
{ x: 2.12, y: 2.12, z: 0 },
{ x: -2.12, y: 2.12, z: 0 },
],
color: getColor("AMBER"),
labels: [{ text: "chord", at: 0, offset: [0, 0.5, 0] }],
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

**Important elements of circle:**

- **Center of circle (O)**: Fixed point that becomes the reference of the circle
- **Radius (r)**: Distance from center to any point on the circle
- **Diameter (d)**: Chord that passes through the center of the circle, $$d = 2r$$
- **Chord**: Line segment that connects two points on the circle

Visible text: - **Center of circle (O)**: Fixed point that becomes the reference of the circle
- **Radius (r)**: Distance from center to any point on the circle
- **Diameter (d)**: Chord that passes through the center of the circle, 
- **Chord**: Line segment that connects two points on the circle

## Arc of Circle

Arc of circle is a part of the circumference of the circle that is bounded by two points on the circle. Arc is denoted with a curved symbol above the letters, for example $$\overset{\frown}{AB}$$.

Visible text: Arc of circle is a part of the circumference of the circle that is bounded by two points on the circle. Arc is denoted with a curved symbol above the letters, for example .

### Types of Arc

Component: LineEquation
Props:
- title: Minor Arc and Major Arc
- description: Visualization of the difference between minor arc and major arc.
- data: [
{
points: Array.from({ length: 91 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("VIOLET"),
showPoints: false,
labels: [{ text: "Minor arc", at: 45, offset: [1, 1, 0] }],
},
{
points: [
{ x: 0, y: 3, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("TEAL"),
showPoints: true,
},
{
points: Array.from({ length: 271 }, (_, i) => {
const angle = ((i + 90) * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("TEAL"),
showPoints: false,
labels: [
{
text: "Major arc",
at: 135,
offset: [-1, -1, 0],
},
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 3, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

**Types of arc based on their length:**

- **Minor arc**: Arc whose length is less than half the circumference of the circle
- **Major arc**: Arc whose length is more than half the circumference of the circle
- **Semicircle**: Arc whose length is exactly half the circumference of the circle

## Central Angle and Inscribed Angle

### Central Angle

Central angle is an angle whose vertex is at the center of the circle and whose sides are radii of the circle.

Component: LineEquation
Props:
- title: Central Angle $$\angle AOB$$
  Visible text: Central Angle
- description: Angle formed by two radii with vertex at the center of the circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2.12, y: 2.12, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
},
{
points: Array.from({ length: 46 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: "Arc AB", at: 22, offset: [1.5, 0.5, 0] }],
},
{
points: Array.from({ length: 46 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 0.8 * Math.cos(angle),
y: 0.8 * Math.sin(angle),
z: 0,
};
}),
color: getColor("CYAN"),
showPoints: false,
labels: [{ text: "α", at: 22, offset: [0.3, 0.2, 0] }],
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

**Properties of central angle:**

- The measure of central angle equals the measure of the arc it subtends
- If central angle is $$\alpha$$, then arc = $$\alpha$$

Visible text: - The measure of central angle equals the measure of the arc it subtends
- If central angle is , then arc =

### Inscribed Angle

Inscribed angle is an angle whose vertex is on the circle and whose sides are chords.

Component: LineEquation
Props:
- title: Inscribed Angle $$\angle ACB$$
  Visible text: Inscribed Angle
- description: Angle whose vertex is on the circle and subtends the same arc.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 3, y: 0, z: 0 },
{ x: -2.12, y: -2.12, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.5, 0, 0] },
{ text: "C", at: 1, offset: [-0.5, -0.3, 0] },
],
},
{
points: [
{ x: 2.12, y: 2.12, z: 0 },
{ x: -2.12, y: -2.12, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 0, offset: [0.3, 0.3, 0] }],
},
{
points: Array.from({ length: 46 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: "Arc AB", at: 22, offset: [1.5, 0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2.12, y: 2.12, z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
labels: [{ text: "O", at: 0, offset: [1, 0.5, 0] }],
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

## Relationship Between Central Angle and Inscribed Angle

Component: MathContainer
Children:

```math
\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}
```

```math
\angle ACB = \frac{1}{2} \times \angle AOB
```

If central angle and inscribed angle subtend the same arc, then the measure of inscribed angle is half the measure of central angle.

**Example application:**

Component: LineEquation
Props:
- title: Relationship Between Central Angle and Inscribed Angle
- description: If $$\angle AOB = 80^\circ$$, then{" "}
$$\angle ACB = 40^\circ$$.
  Visible text: If , then{" "}
.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0.78, y: 2.9, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0, 0.5, 0] }],
},
{
points: [
{ x: 3, y: 0, z: 0 },
{ x: -2.12, y: -2.12, z: 0 },
],
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
},
{
points: [
{ x: 0.78, y: 2.9, z: 0 },
{ x: -2.12, y: -2.12, z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
},
{
points: Array.from({ length: 81 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("AMBER"),
showPoints: false,
},
{
points: Array.from({ length: 20 }, (_, i) => {
const angle = (i * 4 * Math.PI) / 180;
return {
x: 0.8 * Math.cos(angle),
y: 0. ... [truncated; 1340 chars]
- cameraPosition: [0, 0, 10]
- showZAxis: false

## Arc Length and Sector Area

### Arc Length

Arc length is directly proportional to the measure of central angle that subtends it.

Component: MathContainer
Children:

```math
\text{Arc length} = \frac{\alpha}{360^\circ} \times 2\pi r
```

```math
l = \frac{\alpha}{360^\circ} \times 2\pi r
```

Where:

- $$l$$ = arc length
- $$\alpha$$ = measure of central angle (in degrees)
- $$r$$ = radius of circle

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

### Sector Area

Sector is the region bounded by two radii and an arc of circle.

Component: MathContainer
Children:

```math
\text{Sector area} = \frac{\alpha}{360^\circ} \times \pi r^2
```

```math
L = \frac{\alpha}{360^\circ} \times \pi r^2
```

We can visualize the sector area using the equation above.

Component: LineEquation
Props:
- title: Circle Sector
- description: Region bounded by two radii and an arc.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1.5, y: 2.6, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
},
{
points: Array.from({ length: 61 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("AMBER"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
...Array.from({ length: 61 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
{ x: 0, y: 0, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: "Sector", at: 30, offset: [0, 0, 0] }],
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

## Calculating Arc Length and Sector Area

A circle has a radius of $$14 \text{ cm}$$. If the central angle that subtends an arc is $$90^\circ$$, determine:

Visible text: A circle has a radius of . If the central angle that subtends an arc is , determine:

1. Arc length
2. Sector area

**Solution:**

Given: $$r = 14 \text{ cm}$$, $$\alpha = 90^\circ$$

Visible text: Given: ,

1. Arc length:

   <MathContainer>
     
   
   ```math
   l = \frac{\alpha}{360^\circ} \times 2\pi r
   ```

     
   
   ```math
   l = \frac{90^\circ}{360^\circ} \times 2\pi \times 14
   ```

     
   
   ```math
   l = \frac{1}{4} \times 28\pi
   ```

     
   
   ```math
   l = 7\pi \text{ cm} \approx 21.99 \text{ cm}
   ```

   </MathContainer>

2. Sector area:

   <MathContainer>
     
   
   ```math
   L = \frac{\alpha}{360^\circ} \times \pi r^2
   ```

     
   
   ```math
   L = \frac{90^\circ}{360^\circ} \times \pi \times 14^2
   ```

     
   
   ```math
   L = \frac{1}{4} \times 196\pi
   ```

     
   
   ```math
   L = 49\pi \text{ cm}^2 \approx 153.94 \text{ cm}^2
   ```

   </MathContainer>

Visible text: 1. Arc length:

 <MathContainer>
 
 

 
 

 
 

 
 

 </MathContainer>

2. Sector area:

 <MathContainer>
 
 

 
 

 
 

 
 

 </MathContainer>

## Practice Problems

1. A circle has a diameter of $$20 \text{ cm}$$. If an inscribed angle that subtends an arc is $$30^\circ$$, determine the measure of central angle that subtends the same arc!

2. In a circle with center $$O$$ and radius $$21 \text{ cm}$$, there is an arc $$AB$$ with central angle $$120^\circ$$. Calculate:

   - Arc length $$AB$$
   - Sector area $$AOB$$

3. Two inscribed angles subtend the same arc. If one inscribed angle measures $$45^\circ$$, determine the measure of the other inscribed angle!

Visible text: 1. A circle has a diameter of . If an inscribed angle that subtends an arc is , determine the measure of central angle that subtends the same arc!

2. In a circle with center and radius , there is an arc with central angle . Calculate:

 - Arc length 
 - Sector area 

3. Two inscribed angles subtend the same arc. If one inscribed angle measures , determine the measure of the other inscribed angle!

### Answer Key

1. $$\text{Central angle} = 2 \times \text{inscribed angle} = 2 \times 30^\circ = 60^\circ$$

2. Given: $$r = 21 \text{ cm}$$, $$\alpha = 120^\circ$$

Visible text: 1. 

2. Given: ,

- Arc length $$AB = \frac{120^\circ}{360^\circ} \times 2\pi \times 21 = \frac{1}{3} \times 42\pi = 14\pi \text{ cm} \approx 43.98 \text{ cm}$$
- Sector area $$AOB = \frac{120^\circ}{360^\circ} \times \pi \times 21^2 = \frac{1}{3} \times 441\pi = 147\pi \text{ cm}^2 \approx 461.81 \text{ cm}^2$$

Visible text: - Arc length 
- Sector area

3. Inscribed angles that subtend the same arc have the same measure, so the other inscribed angle is $$45^\circ$$

Visible text: 3. Inscribed angles that subtend the same arc have the same measure, so the other inscribed angle is