# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Sector",
  description: "Learn circle sectors, their components, and central angle relationships. Explore minor and major sectors with applications in architecture and engineering.",
  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";

## Understanding Sectors

A sector is an area within a circle bounded by two radii and one arc. Imagine a slice of pizza or cake cut from the center to the edge of the circle. The area formed is what we call a sector.

Every sector has three main components that are interconnected. First are the two radii that meet at the center of the circle. Second is the arc that connects the endpoints of both radii. Third is the area enclosed by both radii and the arc.

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## Characteristics of Sectors

Sectors have special properties that distinguish them from other circle elements. A sector always forms a slice shape with one vertex point at the center of the circle. This point becomes the meeting place of both radii that form the central angle.

The size of a sector is determined by the central angle formed by the two radii. The larger the central angle, the wider the sector area formed. Conversely, a small central angle will produce a narrow sector.

Every sector also has an arc that forms its curved boundary. The length of this arc is proportional to the size of the central angle. In one circle, sectors with the same central angle will have the same area.

## Relationship Between Central Angle and Sector

The central angle is the main determining factor in sector formation. This angle is measured from the center of the circle and formed by two radii that bound the sector. Measurement units can use degrees or radians.

When the central angle increases, the sector will become wider. For example, a sector with a central angle of <InlineMath math="90°" /> will have an area equal to one-quarter of the total circle area. Meanwhile, a sector with a central angle of <InlineMath math="180°" /> will have an area equal to half the circle.

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The mathematical relationship between central angle and sector can be expressed in a simple ratio. If the central angle is <InlineMath math="θ" /> in degrees, then the ratio of sector area to circle area is <InlineMath math="\frac{θ}{360°}" />. For angles in radians, the ratio becomes <InlineMath math="\frac{θ}{2π}" />.

## Applications of Sectors in Life

The concept of sectors is widely applied in various practical fields. In architecture, sectors are used to design curved roofs, arched windows, or circular decorative elements. Sector shapes provide an elegant and proportional impression to buildings.

In engineering, sectors become the basis for calculations in gears, fans, and rotating machine components. Engineers use sector principles to determine load distribution and rotational efficiency.

In daily life, we often encounter sectors on wall clocks. Each hour number divides the circle into 12 equal sectors, each with a central angle of <InlineMath math="30°" />. Pizza slices or cake pieces are also real examples of sector applications in culinary arts.