# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/circle-arc-sector/circle-sector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/circle-arc-sector/circle-sector/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Circle Sector",
  description: "Calculate sector areas using proportional formulas. Understand central angle relationships and solve practical sector area problems with step-by-step methods.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Circle Arcs and Sectors",
};

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

## Definition of Circle Sector

A circle sector is an area bounded by two radii of a circle and an arc of the circle. Imagine a wide banana leaf, then we cut it from the base to the tip following the curved shape of the leaf. Each piece formed resembles the shape of a circle sector.

A sector consists of three main components:

- Two radii that meet at the center of the circle
- One arc that connects the ends of both radii

In geometric form, a sector looks like an orange slice or a curved eyeglass segment. Each sector has a central angle that determines the size of the sector area.

## Relationship Between Central Angle and Sector Area

The area of a sector greatly depends on the size of the central angle that forms it. The central angle is the angle formed by two radii at the center of the circle. The larger the central angle, the larger the sector area formed.

The relationship between central angle and sector area can be expressed as a ratio. If the central angle is <InlineMath math="\alpha" /> degrees, then the sector area is <InlineMath math="\frac{\alpha}{360°}" /> part of the total circle area.

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

This formula shows that the sector area is proportional to its central angle. When the central angle is 90°, the sector area will be one-quarter of the circle area. Similarly, when the central angle is 180°, the sector area will be half of the circle area.

## Calculating Sector Area

To calculate the sector area, we need two important pieces of information: the circle radius and the central angle size. The calculation process is quite simple using the formula we have learned.

Let's look at a calculation example. Suppose there is a circle with radius 14 cm and central angle 60°. The sector area can be calculated as follows:

<div className="flex flex-col gap-4">
<BlockMath math="\text{Sector Area} = \frac{60°}{360°} \times \pi \times 14^2" />
<BlockMath math="= \frac{1}{6} \times \frac{22}{7} \times 196" />
<BlockMath math="= \frac{22 \times 196}{6 \times 7}" />
<BlockMath math="= \frac{4312}{42}" />
<BlockMath math="= \frac{2156}{21} = 102\frac{14}{21} = 102\frac{2}{3} \text{ cm}^2" />
</div>

This calculation shows that a sector with a central angle of 60° has an area of <InlineMath math="102\frac{2}{3}" /> cm².

## Comparison of Central Angle with Sector Area

To understand the proportional relationship between central angle and sector area, let's look at various comparison examples:

<div className="flex flex-col gap-4">

<LineEquation
  title={<>Sector <InlineMath math="60°" /> (<InlineMath math="\frac{1}{6}" /> of circle)</>}
  description={<>Central angle <InlineMath math="60°" /> produces a sector with area <InlineMath math="\frac{1}{6}" /> of the total circle area.</>}
  cameraPosition={[6, 4, 6]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 61 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 60;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("INDIGO"),
      showPoints: false,
      smooth: true,
      lineWidth: 1,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3 * Math.cos(Math.PI / 3), y: 3 * Math.sin(Math.PI / 3), z: 0 },
      ],
      color: getColor("ORANGE"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const angle = (i * Math.PI / 3) / 20;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("ROSE"),
      showPoints: false,
      smooth: true,
      lineWidth: 4,
      labels: [
        {
          text: "60° = 1/6",
          at: 10,
          offset: [0.8, 0.5, 0],
          color: getColor("VIOLET"),
        },
      ],
    },
  ]}
/>

<LineEquation
  title={<>Sector <InlineMath math="90°" /> (<InlineMath math="\frac{1}{4}" /> of circle)</>}
  description={<>Central angle <InlineMath math="90°" /> produces a sector with area <InlineMath math="\frac{1}{4}" /> of the total circle area.</>}
  cameraPosition={[6, 4, 6]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 61 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 60;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("INDIGO"),
      showPoints: false,
      smooth: true,
      lineWidth: 1,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("EMERALD"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 0, y: 3, z: 0 },
      ],
      color: getColor("EMERALD"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const angle = (i * Math.PI / 2) / 30;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("LIME"),
      showPoints: false,
      smooth: true,
      lineWidth: 4,
      labels: [
        {
          text: "90° = 1/4",
          at: 15,
          offset: [0.5, 0.8, 0],
          color: getColor("VIOLET"),
        },
      ],
    },
  ]}
/>

<LineEquation
  title={<>Sector <InlineMath math="120°" /> (<InlineMath math="\frac{1}{3}" /> of circle)</>}
  description={<>Central angle <InlineMath math="120°" /> produces a sector with area <InlineMath math="\frac{1}{3}" /> of the total circle area.</>}
  cameraPosition={[6, 4, 6]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 61 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 60;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("INDIGO"),
      showPoints: false,
      smooth: true,
      lineWidth: 1,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3 * Math.cos(2 * Math.PI / 3), y: 3 * Math.sin(2 * Math.PI / 3), z: 0 },
      ],
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const angle = (i * 2 * Math.PI / 3) / 40;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("CYAN"),
      showPoints: false,
      smooth: true,
      lineWidth: 4,
      labels: [
        {
          text: "120° = 1/3",
          at: 20,
          offset: [-0.5, 1, 0],
          color: getColor("VIOLET"),
        },
      ],
    },
  ]}
/>

<LineEquation
  title={<>Sector <InlineMath math="180°" /> (<InlineMath math="\frac{1}{2}" /> of circle)</>}
  description={<>Central angle <InlineMath math="180°" /> produces a sector with area <InlineMath math="\frac{1}{2}" /> of the total circle area.</>}
  cameraPosition={[6, 4, 6]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 61 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 60;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("INDIGO"),
      showPoints: false,
      smooth: true,
      lineWidth: 1,
    },
    {
      points: [
        { x: -3, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("TEAL"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const angle = (i * Math.PI) / 30;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      smooth: true,
      lineWidth: 4,
      labels: [
        {
          text: "180° = 1/2",
          at: 15,
          offset: [0, 0.5, 0],
          color: getColor("VIOLET"),
        },
      ],
    },
  ]}
/>

<LineEquation
  title={<>Sector <InlineMath math="270°" /> (<InlineMath math="\frac{3}{4}" /> of circle)</>}
  description={<>Central angle <InlineMath math="270°" /> produces a sector with area <InlineMath math="\frac{3}{4}" /> of the total circle area.</>}
  cameraPosition={[6, 4, 6]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 61 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 60;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("INDIGO"),
      showPoints: false,
      smooth: true,
      lineWidth: 1,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("FUCHSIA"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 0, y: -3, z: 0 },
      ],
      color: getColor("FUCHSIA"),
      showPoints: false,
      smooth: false,
      lineWidth: 2,
    },
    {
      points: Array.from({ length: 82 }, (_, i) => {
        const startAngle = 0;
        const endAngle = (3 * Math.PI) / 2;
        const angle = startAngle + (i * endAngle) / 81;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("ORANGE"),
      showPoints: false,
      smooth: true,
      lineWidth: 4,
      labels: [
        {
          text: "270° = 3/4",
          at: 40,
          offset: [1, -0.5, 0],
          color: getColor("VIOLET"),
        },
      ],
    },
  ]}
/>

</div>

## Practice Problems

1. A circle has a radius of 21 cm. Determine the sector area formed by a central angle of 120°.

2. Given that a sector area is 154 cm² and the circle radius is 14 cm. What is the central angle of the sector?

3. A round bread in the shape of a circle with diameter 28 cm is cut into 8 equal parts. Determine the area of each bread piece.

### Answer Key

1. **Solution Steps:**

   Given: <InlineMath math="r = 21 \text{ cm}" />, <InlineMath math="\alpha = 120°" />
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\text{Sector Area} = \frac{\alpha}{360°} \times \pi r^2" />
   <BlockMath math="= \frac{120°}{360°} \times \frac{22}{7} \times 21^2" />
   <BlockMath math="= \frac{1}{3} \times \frac{22}{7} \times 441" />
   <BlockMath math="= \frac{22 \times 441}{3 \times 7}" />
   <BlockMath math="= \frac{9702}{21}" />
   <BlockMath math="= 462 \text{ cm}^2" />
   </div>

2. **Solution Steps:**

   Given: Sector area = 154 cm², <InlineMath math="r = 14 \text{ cm}" />
   
   <div className="flex flex-col gap-4">
   <BlockMath math="154 = \frac{\alpha}{360°} \times \frac{22}{7} \times 14^2" />
   <BlockMath math="154 = \frac{\alpha}{360°} \times \frac{22}{7} \times 196" />
   <BlockMath math="154 = \frac{\alpha}{360°} \times \frac{22 \times 196}{7}" />
   <BlockMath math="154 = \frac{\alpha}{360°} \times \frac{4312}{7}" />
   <BlockMath math="154 = \frac{\alpha}{360°} \times 616" />
   <BlockMath math="\frac{\alpha}{360°} = \frac{154}{616} = \frac{1}{4}" />
   <BlockMath math="\alpha = \frac{1}{4} \times 360° = 90°" />
   </div>

3. **Solution Steps:**

   Given: diameter = 28 cm, so <InlineMath math="r = 14 \text{ cm}" />
   Bread is cut into 8 equal parts, so central angle of each piece = <InlineMath math="\frac{360°}{8} = 45°" />
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\text{Area of each piece} = \frac{45°}{360°} \times \frac{22}{7} \times 14^2" />
   <BlockMath math="= \frac{1}{8} \times \frac{22}{7} \times 196" />
   <BlockMath math="= \frac{22 \times 196}{8 \times 7}" />
   <BlockMath math="= \frac{4312}{56}" />
   <BlockMath math="= 77 \text{ cm}^2" />
   </div>