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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/segment
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/segment/en.mdx
Understand circle segments bounded by chord and arc. Learn minor and major types, area calculations using sector-triangle methods, and tank applications.
---
## Basic Concept of Segment
A segment is one part of a circle that we often encounter in daily life. Imagine a round cake that is cut with a straight knife, the part that is separated from the cake is similar to the concept of a segment in circle geometry.
A segment is a region inside a circle that is bounded by a chord and the arc of the circle that is in front of the chord. In other words, a segment is the part of a circle that is "cut off" by a straight line connecting two points on the circumference of the circle.
## Difference Between Arc and Segment
Start with the fundamental difference between an arc and a segment:
Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Circle Arc
- description: An arc is a curved line on the circumference of a circle that connects two points. An arc only has length in one dimension and has no area.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
{
points: createCircleOutlinePoints(4),
color: getColor("INDIGO"),
showPoints: false,
smooth: true,
lineWidth: 1,
},
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
color: getColor("VIOLET"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "Chord",
at: 1,
offset: [0, -0.8, 0],
},
],
},
createCircleArcLine({
color: getColor("ORANGE"),
lineWidth: 4,
label: {
text: "Arc (curved line)",
offset: [0, 0.8, 0],
},
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
]
Component: LineEquation
Props:
- title: Circle Segment
- description: A segment is a region (area) inside a circle that is bounded by an arc and a chord. A segment has area in two dimensions and can be filled with color.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
{
points: createCircleOutlinePoints(4),
color: getColor("INDIGO"),
showPoints: false,
smooth: true,
lineWidth: 1,
},
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
color: getColor("VIOLET"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "Chord",
at: 1,
offset: [0, -0.8, 0],
},
],
},
createCircleArcLine({
color: getColor("ORANGE"),
lineWidth: 2,
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
...createCircleSegmentBoundaryLines({
color: getColor("LIME"),
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
label: {
text: "Segment (area/region)",
offset: [0, 0.8, 0],
},
}),
]
**Summary of Differences:**
- **Arc**: Curved line on the circumference of a circle (only has length, no area)
- **Segment**: Region surrounded by arc and chord (has area, can be calculated)
## Types of Segment Based on Size
Based on their size, segments can be divided into two types with different characteristics:
Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Minor Segment (Central Angle < $$180^\circ$$)
Visible text: Minor Segment (Central Angle < )
- description: A segment whose area is smaller than half a circle, formed when the central angle is less than $$180^\circ$$. This example shows a segment with a central angle of $$120^\circ$$.
Visible text: A segment whose area is smaller than half a circle, formed when the central angle is less than . This example shows a segment with a central angle of .
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
// Main circle
{
points: createCircleOutlinePoints(4),
color: getColor("INDIGO"),
showPoints: false,
smooth: true,
lineWidth: 1,
},
// Center point O
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("ROSE"),
showPoints: true,
smooth: false,
lineWidth: 3,
labels: [
{
text: "O",
at: 0,
offset: [-0.4, -0.4, 0],
},
],
},
// Radius OA
{
points: createCircleRadiusPoints({ radius: 4, degrees: 30 }),
color: getColor("CYAN"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "A",
at: 1,
offset: [0.4, 0.2, 0],
},
],
},
// Radius OB
{
points: createCircleRadiusPoints({ radius: 4, degrees: 150 }),
color: getColor("CYAN"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "B",
at: 1,
offset: [-0.4, 0.2, 0],
},
],
},
// Central angle arc to show 120°
createCircleArcLine({
color: getColor("AMBER"),
lineWidth: 3,
label: {
text: "120°",
offset: [0, -0.5, 0],
},
radius: 1.5,
startDegrees: 30,
sweepDegrees: 120,
}),
// Chord AB
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
}),
color: getColor("PURPLE"),
showPoints: false,
smooth: false,
lineWidth: 3,
},
// Segment (shaded area)
...createCircleSegmentBoundaryLines({
color: getCol ... [truncated; 1323 chars]
Component: LineEquation
Props:
- title: Major Segment (Central Angle > $$180^\circ$$)
Visible text: Major Segment (Central Angle > )
- description: A segment whose area is larger than half a circle, formed when the central angle is more than $$180^\circ$$. This example shows a segment with a central angle of $$240^\circ$$.
Visible text: A segment whose area is larger than half a circle, formed when ... [truncated; 3572 chars]
**Key Differences:**
- **Minor Segment**: central angle $$\lt 180^\circ$$, segment area $$\lt$$ half circle area
- **Major Segment**: central angle $$\gt 180^\circ$$, segment area $$\gt$$ half circle area
Visible text: - **Minor Segment**: central angle , segment area half circle area
- **Major Segment**: central angle , segment area half circle area
## Segment Area Formula
To calculate the area of a segment, we need to understand that a segment is formed from a sector minus the triangle formed by two radii and a chord.
**Why is it different from the arc length formula?**
- **Arc Length** is $$\frac{\alpha}{360^\circ} \times 2\pi r$$ (units: cm, m, etc.)
- **Segment Area** $$\neq$$ Arc Length (because segment is a $$2\text{D}$$ area, not a $$1\text{D}$$ line)
```math
\text{Segment Area} = \text{Sector Area} - \text{Triangle Area}
```
In more detail, the formula can be written as:
```math
\text{Segment Area} = \frac{\alpha}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin \alpha
```
Where $$\alpha$$ is the central angle in degrees and $$r$$ is the radius of the circle.
Visible text: - **Arc Length** is (units: cm, m, etc.)
- **Segment Area** Arc Length (because segment is a area, not a line)
In more detail, the formula can be written as:
Where is the central angle in degrees and is the radius of the circle.
**Note the difference in units:**
- Arc length: $$\text{cm}$$, $$\text{m}$$ (length units)
- Segment area: $$\text{cm}^2$$, $$\text{m}^2$$ (area units)
Visible text: - Arc length: , (length units)
- Segment area: , (area units)
## Visualization of Segment Formation
Let's see how a segment is formed from its components:
Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Components of Segment Formation
- description: A segment is formed from sector $$AOB$$ minus triangle{" "}
$$AOB$$, with a central angle of{" "}
$$90^\circ$$ and radius of{" "}
$$4 \text{ units}$$.
Visible text: A segment is formed from sector minus triangle{" "}
, with a central angle of{" "}
and radius of{" "}
.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
{
points: createCircleOutlinePoints(4),
color: getColor("INDIGO"),
showPoints: false,
smooth: true,
lineWidth: 1,
},
{
points: createCircleRadiusPoints({ radius: 4, degrees: 0 }),
color: getColor("VIOLET"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "O",
at: 0,
offset: [-0.3, -0.3, 0],
},
{
text: "A",
at: 1,
offset: [0.3, -0.3, 0],
},
],
},
{
points: createCircleRadiusPoints({ radius: 4, degrees: 90 }),
color: getColor("VIOLET"),
showPoints: false,
smooth: false,
lineWidth: 2,
labels: [
{
text: "B",
at: 1,
offset: [-0.3, 0.3, 0],
},
],
},
{
points: createCircleChordPoints({
radius: 4,
startDegrees: 0,
sweepDegrees: 90,
}),
color: getColor("LIME"),
showPoints: false,
smooth: false,
lineWidth: 3,
labels: [
{
text: "Chord AB",
at: 1,
offset: [2.5, 0.5, 0],
},
],
},
createCircleArcLine({
color: getColor("FUCHSIA"),
lineWidth: 4,
label: {
text: "Arc AB (90°)",
offset: [2, 0.5, 0],
},
radius: 4,
startDegrees: 0,
sweepDegrees: 90,
}),
]
## Segment Area Calculation
To understand how to calculate the area of a segment, let's use an example with a central angle of $$90^\circ$$ and radius $$r = 10 \text{ cm}$$, according to the visualization of segment formation components above.
Visible text: To understand how to calculate the area of a segment, let's use an example with a central angle of and radius , according to the visualization of segment formation components above.
**Step** $$1$$: Calculate Sector Area
Visible text: **Step** : Calculate Sector Area
Component: MathContainer
Children:
```math
\text{Sector Area} = \frac{90^\circ}{360^\circ} \times \pi r^2
```
```math
= \frac{1}{4} \times \pi \times 10^2
```
```math
= \frac{1}{4} \times \pi \times 100
```
```math
= 25\pi \text{ cm}^2 = 78.5398 \text{ cm}^2
```
**Step** $$2$$: Calculate Triangle Area
Visible text: **Step** : Calculate Triangle Area
For a $$90^\circ$$ angle, the triangle formed is a right triangle with both perpendicular sides being radii:
Visible text: For a angle, the triangle formed is a right triangle with both perpendicular sides being radii:
Component: MathContainer
Children:
```math
\text{Triangle Area} = \frac{1}{2} \times r^2 \times \sin 90^\circ
```
```math
= \frac{1}{2} \times 10^2 \times 1
```
```math
= \frac{1}{2} \times 100 \times 1
```
```math
= 50 \text{ cm}^2
```
**Step** $$3$$: Calculate Segment Area
Visible text: **Step** : Calculate Segment Area
Component: MathContainer
Children:
```math
\text{Segment Area} = \text{Sector Area} - \text{Triangle Area}
```
```math
= 25\pi - 50
```
```math
= 78.5398 - 50
```
```math
= 28.5398 \text{ cm}^2
```
## Exercises
1. A circle has a radius of $$7 \text{ cm}$$. If there is a segment with a central angle of $$60^\circ$$, determine the area of the segment.
2. A segment is known to have an area of $$15.7 \text{ cm}^2$$ on a circle with a radius of $$5 \text{ cm}$$. Determine the central angle that forms the segment.
3. A cylindrical tank with a radius of $$2 \text{ meters}$$ contains water up to a depth of $$1.5 \text{ meters}$$ from the bottom of the tank. Determine the area of the water surface visible from above.
Visible text: 1. A circle has a radius of . If there is a segment with a central angle of , determine the area of the segment.
2. A segment is known to have an area of on a circle with a radius of . Determine the central angle that forms the segment.
3. A cylindrical tank with a radius of contains water up to a depth of from the bottom of the tank. Determine the area of the water surface visible from above.
### Answer Key
1. **Solution Steps:**
Given: $$r = 7 \text{ cm}$$, $$\alpha = 60^\circ$$
Calculate sector area:
```math
\text{Sector Area} = \frac{60^\circ}{360^\circ} \times \pi \times 7^2
```
```math
= \frac{1}{6} \times \pi \times 49
```
```math
= \frac{49\pi}{6} \text{ cm}^2
```
Calculate triangle area:
```math
\text{Triangle Area} = \frac{1}{2} r^2 \sin 60^\circ
```
```math
= \frac{1}{2} \times 49 \times \frac{\sqrt{3}}{2}
```
```math
= \frac{49\sqrt{3}}{4} \text{ cm}^2
```
Calculate segment area:
```math
\text{Segment Area} = \frac{49\pi}{6} - \frac{49\sqrt{3}}{4}
```
```math
= 25.6563 - 21.2176 = 4.4387 \text{ cm}^2
```
2. **Solution Steps:**
Given: segment area = $$15.7 \text{ cm}^2$$, $$r = 5 \text{ cm}$$
Using the formula: $$\text{Segment Area} = \frac{\alpha}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin \alpha$$
Using trial and numerical approximation or graphing calculator, we get:
$$\alpha \approx 120^\circ$$
Verification:
```math
\text{Sector Area} = \frac{120^\circ}{360^\circ} \times \pi \times 25 = \frac{25\pi}{3}
```
```math
\text{Triangle Area} = \frac{1}{2} \times 25 \times \sin 120^\circ = \frac{25\sqrt{3}}{4}
```
```math
\text{Segment Area} = \frac{25\pi}{3} - \frac{25\sqrt{3}}{4} \approx 26.1799 - 10.8253 = 15.3546 \text{ cm}^2
```
The calculated result ($$15.3546 \text{ cm}^2$$) is slightly different from what is given in the problem ($$15.7 \text{ cm}^2$$). This difference is due to the precision of the values $$\pi$$ and $$\sqrt{3}$$ used. Mathematically, the calculated result $$15.3546 \text{ cm}^2$$ is the most accurate.
3. **Solution Steps:**
Given: tank radius = $$2 \text{ m}$$, water depth = $$1.5 \text{ m}$$
Since the water depth ($$1.5 \text{ m}$$) is less than the radius ($$2 \text{ m}$$), the water surface forms a segment.
For horizontal cylindrical tanks, we use a special segment formula with a specific height.
Determine segment height from tank bottom:
```math
h = 1.5 \text{ m (water height from bottom)}
```
Using the segment area formula for water segment in horizontal tanks:
```math
\text{Area} = r^2 \times \left[\arccos\left(\frac{r-h}{r}\right) - \frac{r-h}{r} \times \sqrt{1-\left(\frac{r-h}{r}\right)^2}\right]
```
Substitute values:
```math
\frac{r-h}{r} = \frac{2-1.5}{2} = 0.25
```
```math
\arccos(0.25) = 1.3181 \text{ radians}
```
```math
\sqrt{1-0.25^2} = \sqrt{0.9375} = 0.9682
```
Calculate water surface area:
```math
\text{Area} = 4 \times [1.3181 - 0.25 \times 0.9682]
```
```math
= 4 \times [1.3181 - 0.2421]
```
```math
= 4 \times 1.0760 = 4.3040 \text{ m}^2
```
Visible text: 1. **Solution Steps:**
Given: ,
Calculate sector area:
Calculate triangle area:
Calculate segment area:
2. **Solution Steps:**
Given: segment area = ,
Using the formula:
Using trial and numerical approximation or graphing calculator, we get:
Verification:
The calculated result () is slightly different from what is given in the problem (). This difference is due to the precision of the values and used. Mathematically, the calculated result is the most accurate.
3. **Solution Steps:**
Given: tank radius = , water depth =
Since the water depth () is less than the radius (), the water surface forms a segment.
For horizontal cylindrical tanks, we use a special segment formula with a specific height.
Determine segment height from tank bottom:
Using the segment area formula for water segment in horizontal tanks:
Substitute values:
Calculate water surface area: