# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/circle/circle-and-arc-circle
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/circle/circle-and-arc-circle/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Circle and Arc Circle",
  description: "Learn circle fundamentals, arc types, and sector calculations. Master equations, elements, and solve arc length problems with visual examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Circle",
};

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

## 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 <InlineMath math="O(a,b)" /> and radius <InlineMath math="r" /> can be expressed with the equation:

<BlockMath math="(x-a)^2 + (y-b)^2 = r^2" />

### Elements of Circle

How is the visualization of the circle equation?

<LineEquation
  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, <InlineMath math="d = 2r" />
- **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 <InlineMath math="\overset{\frown}{AB}" />.

### Types of Arc

<LineEquation
  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.

<LineEquation
  title={
    <>
      Central Angle <InlineMath math="\angle AOB" />
    </>
  }
  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 = <InlineMath math="\alpha" />, then arc = <InlineMath math="\alpha" />

### Inscribed Angle

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

<LineEquation
  title={
    <>
      Inscribed Angle <InlineMath math="\angle ACB" />
    </>
  }
  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

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}" />
  <BlockMath math="\angle ACB = \frac{1}{2} \times \angle AOB" />
</div>

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:**

<LineEquation
  title="Relationship Between Central Angle and Inscribed Angle"
  description={
    <>
      If <InlineMath math="\angle AOB = 80°" />, then{" "}
      <InlineMath math="\angle ACB = 40°" />.
    </>
  }
  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: 0.8 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "80°", at: 10, offset: [0.5, 0.2, 0] }],
    },
  ]}
  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.

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Arc length} = \frac{\alpha}{360°} \times 2\pi r" />
  <BlockMath math="l = \frac{\alpha}{360°} \times 2\pi r" />
</div>

Where:

- <InlineMath math="l" /> = arc length
- <InlineMath math="\alpha" /> = measure of central angle (in degrees)
- <InlineMath math="r" /> = radius of circle

### Sector Area

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

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Sector area} = \frac{\alpha}{360°} \times \pi r^2" />
  <BlockMath math="L = \frac{\alpha}{360°} \times \pi r^2" />
</div>

We can visualize the sector area using the equation above.

<LineEquation
  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 cm. If the central angle that subtends an arc is 90°, determine:

1. Arc length
2. Sector area

**Solution:**

Given: <InlineMath math="r = 14" /> cm, <InlineMath math="\alpha = 90°" />

1. Arc length:

   <div className="flex flex-col gap-4">
     <BlockMath math="l = \frac{\alpha}{360°} \times 2\pi r" />
     <BlockMath math="l = \frac{90°}{360°} \times 2\pi \times 14" />
     <BlockMath math="l = \frac{1}{4} \times 28\pi" />
     <BlockMath math="l = 7\pi \text{ cm} \approx 21.99 \text{ cm}" />
   </div>

2. Sector area:

   <div className="flex flex-col gap-4">
     <BlockMath math="L = \frac{\alpha}{360°} \times \pi r^2" />
     <BlockMath math="L = \frac{90°}{360°} \times \pi \times 14^2" />
     <BlockMath math="L = \frac{1}{4} \times 196\pi" />
     <BlockMath math="L = 49\pi \text{ cm}^2 \approx 153.94 \text{ cm}^2" />
   </div>

## Practice Problems

1. A circle has a diameter of 20 cm. If an inscribed angle that subtends an arc is 30°, determine the measure of central angle that subtends the same arc!

2. In a circle with center O and radius 21 cm, there is an arc AB with central angle 120°. Calculate:

   - Arc length AB
   - Sector area AOB

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

### Answer Key

1. Central angle = 2 × inscribed angle = 2 × 30° = 60°

2. Given: r = 21 cm, α = 120°

   - Arc length AB = <InlineMath math="\frac{120°}{360°} \times 2\pi \times 21 = \frac{1}{3} \times 42\pi = 14\pi" /> cm ≈ 43.98 cm
   - Sector area AOB = <InlineMath math="\frac{120°}{360°} \times \pi \times 21^2 = \frac{1}{3} \times 441\pi = 147\pi" /> cm² ≈ 461.81 cm²

3. Inscribed angles that subtend the same arc have the same measure, so the other inscribed angle = 45°