# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Central Angle and Inscribed Angle",
  description: "Master central and inscribed angles in circles. Learn the key relationship, theorems, and solve problems with step-by-step examples and proofs.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Circle",
};

## Definition of Central Angle

A central angle is an angle formed by two radii of a circle with the vertex located at the center of the circle. The sides of the central angle are radii that connect the center to points on the circle.

<LineEquation
  title="Central Angle"
  description="An angle whose vertex is located at the center of the circle."
  data={[
    {
      points: Array.from({ length: 361 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 4, 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: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), 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: 1.2 * Math.cos(angle),
          y: 1.2 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("CYAN"),
      showPoints: false,
      labels: [{ text: "α", at: 22, offset: [0.3, 0.2, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

In the figure above:

- Point O is the center of the circle
- OA and OB are radii of the circle
- <InlineMath math="\angle AOB" /> is the central angle
- The measure of the central angle is denoted by <InlineMath math="\alpha" />

## Definition of Inscribed Angle

An inscribed angle is an angle formed by two chords with the vertex located on the circle. The sides of the inscribed angle are chords that connect the vertex to two other points on the circle.

<LineEquation
  title="Inscribed Angle"
  description="An angle whose vertex is located on the circle."
  data={[
    {
      points: Array.from({ length: 361 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
    },
    {
      points: [
        { x: 4, y: 0, z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), 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: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: true,
      labels: [{ text: "B", at: 0, offset: [0.3, 0.3, 0] }],
    },
    {
      points: [{ x: 0, y: 0, z: 0 }],
      color: getColor("CYAN"),
      showPoints: true,
      labels: [{ text: "O", at: 0, offset: [0, -0.5, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

In the figure above:

- Point C is located on the circle
- CA and CB are chords
- <InlineMath math="\angle ACB" /> is the inscribed angle
- Point O is the center of the circle

## Relationship Between Central Angle and Inscribed Angle

Central angles and inscribed angles that subtend the same arc have a special relationship. Let's observe this relationship.

<LineEquation
  title="Relationship Between Central Angle and Inscribed Angle"
  description="Central angle and inscribed angle that subtend the same arc."
  data={[
    {
      points: Array.from({ length: 361 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 4, 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: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: true,
      labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
    },
    {
      points: [
        { x: 4, y: 0, z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: true,
      labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
    },
    {
      points: [
        { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: false,
    },
    {
      points: Array.from({ length: 46 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * 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: 1.2 * Math.cos(angle),
          y: 1.2 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "α", at: 22, offset: [0.3, 0.2, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

### Theorem of Central Angle and Inscribed Angle Relationship

<BlockMath math="\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}" />

If a central angle and an inscribed angle subtend the same arc, then:

- Measure of inscribed angle = <InlineMath math="\frac{1}{2}" /> × measure of central angle
- Measure of central angle = 2 × measure of inscribed angle

## Proof of Central Angle and Inscribed Angle Relationship

Let's prove the relationship between central angle and inscribed angle by constructing auxiliary lines.

<LineEquation
  title="Proof with Auxiliary Lines"
  description="Constructing auxiliary line from C through O to prove the relationship."
  data={[
    {
      points: Array.from({ length: 361 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 4, 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: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: true,
      labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
    },
    {
      points: [
        { x: 4, y: 0, z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: true,
      labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
    },
    {
      points: [
        { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: false,
    },
    {
      points: [
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
        { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
      ],
      color: getColor("TEAL"),
      showPoints: true,
      labels: [{ text: "D", at: 1, offset: [0.8, 0.5, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

**Proof steps:**

1. Construct line CD that passes through point O (center of the circle)
2. Note that OA = OB = OC = OD (radii of the circle)
3. Triangles AOC and BOC are isosceles triangles
4. Let <InlineMath math="\angle ACO = x" /> and <InlineMath math="\angle BCO = y" />
5. Since they are isosceles triangles: <InlineMath math="\angle CAO = x" /> and <InlineMath math="\angle CBO = y" />
6. Exterior angles of triangles: <InlineMath math="\angle AOD = 2x" /> and <InlineMath math="\angle BOD = 2y" />
7. Therefore: <InlineMath math="\angle AOB = 2x + 2y = 2(x + y) = 2 \times \angle ACB" />

## Properties of Central Angle and Inscribed Angle

1. **Inscribed Angle Subtending a Diameter**

   Every inscribed angle that subtends a diameter of a circle measures 90° (right angle).

   <LineEquation
     title="Inscribed Angle Subtending a Diameter"
     description="Inscribed angle subtending a diameter is always 90°."
     data={[
       {
         points: Array.from({ length: 361 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("PURPLE"),
         showPoints: false,
       },
       {
         points: [
           { x: -4, y: 0, z: 0 },
           { x: 4, y: 0, z: 0 },
         ],
         color: getColor("ORANGE"),
         showPoints: true,
         labels: [
           { text: "A", at: 0, offset: [-0.5, 0, 0] },
           { text: "B", at: 1, offset: [0.5, 0, 0] },
         ],
       },
       {
         points: [
           { x: -4, y: 0, z: 0 },
           { x: 0, y: 4, z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: true,
         labels: [{ text: "C", at: 1, offset: [0, 0.5, 0] }],
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 0, y: 4, z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: false,
       },
       {
         points: [{ x: 0, y: 0, z: 0 }],
         color: getColor("AMBER"),
         showPoints: true,
         labels: [{ text: "O", at: 0, offset: [0, -0.5, 0] }],
       },
       {
         points: (() => {
           const C = { x: 0, y: 4 };
           const A = { x: -4, y: 0 };
           const B = { x: 4, y: 0 };
           
           // Calculate angle ACB at point C (should be 90°)
           const angleCA = Math.atan2(A.y - C.y, A.x - C.x);
           const angleCB = Math.atan2(B.y - C.y, B.x - C.x);
           
           return Array.from({ length: 16 }, (_, i) => {
             const t = i / 15;
             const angle = angleCA + t * (angleCB - angleCA);
             return {
               x: C.x + 0.8 * Math.cos(angle),
               y: C.y + 0.8 * Math.sin(angle),
               z: 0,
             };
           });
         })(),
         color: getColor("PINK"),
         showPoints: false,
         labels: [{ text: "90°", at: 7, offset: [0.5, -0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

2. **Inscribed Angles Subtending the Same Arc**

   <InlineMath math="\angle ACB = \angle ADB" />, both angles subtend the same
   arc AB.

   <LineEquation
     title="Inscribed Angles Subtending the Same Arc"
     description="Inscribed angles that subtend the same arc have equal measures."
     data={[
       {
         points: Array.from({ length: 361 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("PURPLE"),
         showPoints: false,
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
         ],
         color: getColor("ORANGE"),
         showPoints: true,
         labels: [
           { text: "A", at: 0, offset: [0.5, 0, 0] },
           { text: "B", at: 1, offset: [0.3, 0.3, 0] },
         ],
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: true,
         labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
       },
       {
         points: [
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
           { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: false,
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("TEAL"),
         showPoints: true,
         labels: [{ text: "D", at: 1, offset: [-0.5, 0.3, 0] }],
       },
       {
         points: [
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
           { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("TEAL"),
         showPoints: false,
       },
       {
         points: Array.from({ length: 46 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("AMBER"),
         showPoints: false,
         labels: [{ text: "Arc AB", at: 22, offset: [1.5, 0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

## Calculating Inscribed Angle

Given central angle AOB = 80°. Find the measure of inscribed angle ACB that subtends the same arc!

<LineEquation
  title="Visualization"
  description="Central angle AOB = 80°, find inscribed angle ACB."
  data={[
    {
      points: Array.from({ length: 361 }, (_, i) => {
        const angle = (i * Math.PI) / 180;
        return {
          x: 4 * Math.cos(angle),
          y: 4 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 4, 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: 4 * Math.cos((80 * Math.PI) / 180), y: 4 * Math.sin((80 * Math.PI) / 180), z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: true,
      labels: [{ text: "B", at: 1, offset: [0, 0.5, 0] }],
    },
    {
      points: [
        { x: 4, y: 0, z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: true,
      labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
    },
    {
      points: [
        { x: 4 * Math.cos((80 * Math.PI) / 180), y: 4 * Math.sin((80 * Math.PI) / 180), z: 0 },
        { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: false,
    },
    {
      points: Array.from({ length: 18 }, (_, i) => {
        const angle = ((i * 80) / 17) * Math.PI / 180;
        return {
          x: 1.2 * Math.cos(angle),
          y: 1.2 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "80°", at: 10, offset: [0.5, 0.3, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="\angle ACB = \frac{1}{2} \times \angle AOB" />
  <BlockMath math="\angle ACB = \frac{1}{2} \times 80°" />
  <BlockMath math="\angle ACB = 40°" />
</div>

## Calculating Central Angle

Given inscribed angle ACB = 35°. Find the measure of central angle AOB that subtends the same arc!

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="\angle AOB = 2 \times \angle ACB" />
  <BlockMath math="\angle AOB = 2 \times 35°" />
  <BlockMath math="\angle AOB = 70°" />
</div>

## Practice Problems

1. If the central angle of a circle is 120°, what is the measure of the inscribed angle that subtends the same arc?

2. Inscribed angle ABC = 45°. Find the measure of central angle AOB!

3. In a circle, inscribed angle PQR subtends a diameter. What is the measure of angle PQR?

4. Two inscribed angles subtend the same arc. If one angle measures 25°, find the measure of the other angle!

### Answer Key

1. Inscribed angle = <InlineMath math="\frac{1}{2} \times 120° = 60°" />

2. Central angle AOB = <InlineMath math="2 \times 45° = 90°" />

3. Angle PQR = 90° (inscribed angle subtending a diameter is always 90°)

4. The other angle = 25° (inscribed angles subtending the same arc have equal measures)
