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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/circle-sector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/circle-sector/en.mdx
Calculate sector areas using proportional formulas. Understand central angle relationships and solve practical sector area problems.
---
## 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 $$\alpha$$ degrees, then the sector area is $$\frac{\alpha}{360^\circ}$$ part of the total circle area.
Visible text: The relationship between central angle and sector area can be expressed as a ratio. If the central angle is degrees, then the sector area is part of the total circle area.
Component: MathContainer
Children:
```math
\text{Sector Area} = \frac{\alpha}{360^\circ} \times \pi r^2
```
This formula shows that the sector area is proportional to its central angle. When the central angle is $$90^\circ$$, the sector area will be one-quarter of the circle area. Similarly, when the central angle is $$180^\circ$$, the sector area will be half of the circle area.
Visible text: This formula shows that the sector area is proportional to its central angle. When the central angle is , the sector area will be one-quarter of the circle area. Similarly, when the central angle is , 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 $$14 \text{ cm}$$ and central angle $$60^\circ$$. The sector area can be calculated as follows:
Visible text: Let's look at a calculation example. Suppose there is a circle with radius and central angle . The sector area can be calculated as follows:
Component: MathContainer
Children:
```math
\text{Sector Area} = \frac{60^\circ}{360^\circ} \times \pi \times 14^2
```
```math
= \frac{1}{6} \times \frac{22}{7} \times 196
```
```math
= \frac{22 \times 196}{6 \times 7}
```
```math
= \frac{4312}{42}
```
```math
= \frac{2156}{21} = 102\frac{14}{21} = 102\frac{2}{3} \text{ cm}^2
```
This calculation shows that a sector with a central angle of $$60^\circ$$ has an area of $$102\frac{2}{3} \text{ cm}^2$$.
Visible text: This calculation shows that a sector with a central angle of has an area of .
## 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:
Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Sector $$60^\circ$$ ($$\frac{1}{6}$$ of circle)
Visible text: Sector ( of circle)
- description: Central angle $$60^\circ$$ produces a sector with area $$\frac{1}{6}$$ of the total circle area.
Visible text: Central angle produces a sector with area 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"),
},
],
},
]
Component: LineEquation
Props:
- title: Sector $$90^\circ$$ ($$\frac{1}{4}$$ of circle)
Visible text: Sector ( of circle)
- description: Central angle $$90^\circ$$ produces a sector with area $$\frac{1}{4}$$ of the total circle area.
Visible text: Central angle produces a sector with area 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) => {
c ... [truncated; 6120 chars]
## Practice Problems
1. A circle has a radius of $$21 \text{ cm}$$. Determine the sector area formed by a central angle of $$120^\circ$$.
2. Given that a sector area is $$154 \text{ cm}^2$$ and the circle radius is $$14 \text{ cm}$$. What is the central angle of the sector?
3. A round bread in the shape of a circle with diameter $$28 \text{ cm}$$ is cut into $$8$$ equal parts. Determine the area of each bread piece.
Visible text: 1. A circle has a radius of . Determine the sector area formed by a central angle of .
2. Given that a sector area is and the circle radius is . What is the central angle of the sector?
3. A round bread in the shape of a circle with diameter is cut into equal parts. Determine the area of each bread piece.
### Answer Key
1. **Solution Steps:**
Given: $$r = 21 \text{ cm}$$, $$\alpha = 120^\circ$$
```math
\text{Sector Area} = \frac{\alpha}{360^\circ} \times \pi r^2
```
```math
= \frac{120^\circ}{360^\circ} \times \frac{22}{7} \times 21^2
```
```math
= \frac{1}{3} \times \frac{22}{7} \times 441
```
```math
= \frac{22 \times 441}{3 \times 7}
```
```math
= \frac{9702}{21}
```
```math
= 462 \text{ cm}^2
```
2. **Solution Steps:**
Given: sector area = $$154 \text{ cm}^2$$, $$r = 14 \text{ cm}$$
```math
154 = \frac{\alpha}{360^\circ} \times \frac{22}{7} \times 14^2
```
```math
154 = \frac{\alpha}{360^\circ} \times \frac{22}{7} \times 196
```
```math
154 = \frac{\alpha}{360^\circ} \times \frac{22 \times 196}{7}
```
```math
154 = \frac{\alpha}{360^\circ} \times \frac{4312}{7}
```
```math
154 = \frac{\alpha}{360^\circ} \times 616
```
```math
\frac{\alpha}{360^\circ} = \frac{154}{616} = \frac{1}{4}
```
```math
\alpha = \frac{1}{4} \times 360^\circ = 90^\circ
```
3. **Solution Steps:**
Given: diameter = $$28 \text{ cm}$$, so $$r = 14 \text{ cm}$$
Bread is cut into $$8$$ equal parts, so central angle of each piece is $$\frac{360^\circ}{8} = 45^\circ$$
```math
\text{Area of each piece} = \frac{45^\circ}{360^\circ} \times \frac{22}{7} \times 14^2
```
```math
= \frac{1}{8} \times \frac{22}{7} \times 196
```
```math
= \frac{22 \times 196}{8 \times 7}
```
```math
= \frac{4312}{56}
```
```math
= 77 \text{ cm}^2
```
Visible text: 1. **Solution Steps:**
Given: ,
2. **Solution Steps:**
Given: sector area = ,
3. **Solution Steps:**
Given: diameter = , so
Bread is cut into equal parts, so central angle of each piece is