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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/arc
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/arc/en.mdx
Learn circle arcs, arc length formulas, and central angle relationships. Learn minor, major, and semicircular arcs with worked examples.
---
## Understanding Circle Arcs
A circle arc is a part of the circumference of a circle that is bounded by two points on the circle. Imagine it like a piece of string that curves following the shape of the circle. When we have a circle and mark two points on it, the part of the circumference that connects these two points is called an arc.
Each arc has two endpoints located on the circle, and the length of the arc depends on the size of the central angle that faces the arc. The larger the central angle, the longer the arc.
Component: LineEquation
Props:
- title: Circle Arc Visualization
- description: Circle arcs with various central angle sizes.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
{
points: createCircleOutlinePoints(4),
color: getColor("CYAN"),
smooth: true,
showPoints: false,
labels: [{ text: "Circle", at: 56, offset: [1.5, 0.5, 0] }]
},
createCircleArcLine({
radius: 4,
startDegrees: 30,
sweepDegrees: 120,
segments: 12,
color: getColor("ORANGE"),
label: { text: "Arc AB", offset: [1.5, 0.5, 0] }
}),
{
points: createCircleRadiusPoints({ radius: 4, degrees: 30 }),
color: getColor("PURPLE"),
smooth: false,
showPoints: true,
labels: [{ text: "A", at: 1, offset: [0.3, 0.3, 0] }]
},
{
points: createCircleRadiusPoints({ radius: 4, degrees: 150 }),
color: getColor("PURPLE"),
smooth: false,
showPoints: true,
labels: [{ text: "B", at: 1, offset: [-0.3, 0.3, 0] }]
}
]
## Types of Arcs Based on Size
Based on the size of the central angle that faces them, circle arcs can be distinguished into several types:
1. **Minor Arc** is an arc whose central angle is less than $$180^\circ$$. This arc is the shorter part of the two possible arcs connecting two points on the circle.
2. **Major Arc** is an arc whose central angle is more than $$180^\circ$$. This arc is the longer part of the two possible arcs connecting two points on the circle.
3. **Semicircular Arc** is an arc whose central angle is exactly $$180^\circ$$. This arc divides the circle into two equal parts.
Visible text: 1. **Minor Arc** is an arc whose central angle is less than . This arc is the shorter part of the two possible arcs connecting two points on the circle.
2. **Major Arc** is an arc whose central angle is more than . This arc is the longer part of the two possible arcs connecting two points on the circle.
3. **Semicircular Arc** is an arc whose central angle is exactly . This arc divides the circle into two equal parts.
## Arc Length Formula
Arc length can be calculated using the ratio between the central angle and the full angle of the circle. Since the full circumference of a circle is $$2\pi r$$, the arc length can be expressed as:
Visible text: Arc length can be calculated using the ratio between the central angle and the full angle of the circle. Since the full circumference of a circle is , the arc length can be expressed as:
```math
\text{Arc length} = \frac{\alpha}{360^\circ} \times 2\pi r
```
where:
- $$\alpha$$ is the central angle in degrees
- $$r$$ is the radius of the circle
Visible text: - is the central angle in degrees
- is the radius of the circle
If the central angle is expressed in radians, the formula becomes simpler:
```math
\text{Arc length} = \alpha \times r
```
where $$\alpha$$ is the central angle in radians.
Visible text: where is the central angle in radians.
Component: LineEquation
Props:
- title: Arc Length Comparison
- description: Arcs with different central angles on the same circle.
- cameraPosition: [0, 0, 10]
- showZAxis: false
- data: [
{
points: createCircleOutlinePoints(3),
color: getColor("CYAN"),
smooth: true,
showPoints: false
},
createCircleArcLine({
radius: 3,
startDegrees: 0,
sweepDegrees: 60,
segments: 6,
color: getColor("ORANGE"),
label: { text: "60°", offset: [0.8, 0.3, 0] }
}),
createCircleArcLine({
radius: 3,
startDegrees: 180,
sweepDegrees: 120,
segments: 12,
color: getColor("PURPLE"),
label: { text: "120°", offset: [-1.5, 0.3, 0] }
})
]
## Relationship Between Arc and Central Angle
There is a very close relationship between arc length and the size of the central angle that faces it. This relationship can be expressed in the form of a ratio:
```math
\frac{\text{Arc length}_1}{\text{Arc length}_2} = \frac{\alpha_1}{\alpha_2}
```
This ratio applies to arcs on the same circle. This means that if the central angle of one arc is twice the central angle of another arc, then the length of that arc will also be twice as long.
This concept is very useful in solving various problems involving circle arcs, especially when we need to find arc length without knowing the radius of the circle directly.
## Practice Problems
1. A circle arc has a radius of $$14 \text{ cm}$$ and a central angle of $$90^\circ$$. Determine the length of the arc.
2. Given that the length of arc $$AB$$ is $$22 \text{ cm}$$ and the angle $$AOB$$ is $$120^\circ$$, where O is the center of the circle. What is the radius of the circle?
3. In a circle with radius $$21 \text{ cm}$$, there are two arcs. The first arc has a central angle of $$60^\circ$$ and the second arc has a central angle of $$150^\circ$$. Determine the ratio of the lengths of the two arcs.
4. A circle arc has a length of $$15.7 \text{ cm}$$. If the radius of the circle is $$10 \text{ cm}$$, determine the central angle of the arc in degrees.
5. Given that the circumference of a circle is $$88 \text{ cm}$$. If an arc on the circle has a central angle of $$45^\circ$$, determine the length of the arc.
Visible text: 1. A circle arc has a radius of and a central angle of . Determine the length of the arc.
2. Given that the length of arc is and the angle is , where O is the center of the circle. What is the radius of the circle?
3. In a circle with radius , there are two arcs. The first arc has a central angle of and the second arc has a central angle of . Determine the ratio of the lengths of the two arcs.
4. A circle arc has a length of . If the radius of the circle is , determine the central angle of the arc in degrees.
5. Given that the circumference of a circle is . If an arc on the circle has a central angle of , determine the length of the arc.
### Answer Key
1. **Answer:** $$22 \text{ cm}$$
Given: $$r = 14 \text{ cm}$$, $$\alpha = 90^\circ$$
Using the arc length formula:
```math
\text{Arc length} = \frac{\alpha}{360^\circ} \times 2\pi r
```
```math
= \frac{90^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 14
```
```math
= \frac{1}{4} \times 2 \times \frac{22}{7} \times 14
```
```math
= \frac{1}{4} \times 88 = 22 \text{ cm}
```
2. **Answer:** $$10.5 \text{ cm}$$
Given: Arc length AB = $$22 \text{ cm}$$, $$\alpha = 120^\circ$$
Using the arc length formula:
```math
22 = \frac{120^\circ}{360^\circ} \times 2\pi r
```
```math
22 = \frac{1}{3} \times 2 \times \frac{22}{7} \times r
```
```math
22 = \frac{44r}{21}
```
```math
r = \frac{22 \times 21}{44} = \frac{462}{44} = 10.5 \text{ cm}
```
3. **Answer:** $$2:5$$
Given: $$r = 21 \text{ cm}$$, $$\alpha_1 = 60^\circ$$, $$\alpha_2 = 150^\circ$$
Since on the same circle, the ratio of arc lengths equals the ratio of their central angles:
```math
\frac{\text{Arc length}_1}{\text{Arc length}_2} = \frac{\alpha_1}{\alpha_2} = \frac{60^\circ}{150^\circ} = \frac{2}{5}
```
Therefore, the ratio of the lengths of the two arcs is $$2:5$$.
4. **Answer:** $$90^\circ$$
Given: arc length = $$15.7 \text{ cm}$$, $$r = 10 \text{ cm}$$
Using the arc length formula:
```math
15.7 = \frac{\alpha}{360^\circ} \times 2\pi \times 10
```
```math
15.7 = \frac{\alpha}{360^\circ} \times 2 \times 3.14 \times 10
```
```math
15.7 = \frac{\alpha}{360^\circ} \times 62.8
```
```math
\alpha = \frac{15.7 \times 360^\circ}{62.8} = \frac{5652^\circ}{62.8} = 90^\circ
```
5. **Answer: $$11 \text{ cm}$$**
Given: Circumference = $$88 \text{ cm}$$, $$\alpha = 45^\circ$$
Using the ratio concept:
```math
\text{Arc length} = \frac{\alpha}{360^\circ} \times \text{Circumference}
```
```math
= \frac{45^\circ}{360^\circ} \times 88
```
```math
= \frac{1}{8} \times 88 = 11 \text{ cm}
```
Visible text: 1. **Answer:**
Given: ,
Using the arc length formula:
2. **Answer:**
Given: Arc length AB = ,
Using the arc length formula:
3. **Answer:**
Given: , ,
Since on the same circle, the ratio of arc lengths equals the ratio of their central angles:
Therefore, the ratio of the lengths of the two arcs is .
4. **Answer:**
Given: arc length = ,
Using the arc length formula:
5. **Answer: **
Given: Circumference = ,
Using the ratio concept: