# Nakafa Learning Content
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URL: https://nakafa.com/en/subjects/mathematics/circle-arc-sector/circle-arc
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle-arc-sector/circle-arc/en.mdx
Learn arc-chord relationships, sagitta calculations, and parametric equations. Apply advanced trigonometric formulas to engineering and architectural designs.
---
## Arc and Chord Relationship
Every circle arc has a close relationship with the chord that connects its two endpoints. A chord is a straight line connecting two endpoints of an arc, while an arc is a curved path along the circle's circumference. Imagine it like a bow and arrow, where the string is the straight line and the bow is the curved wood.
This relationship is very important in various engineering and architectural applications. The longer the arc, the longer the chord that connects it, but this relationship is not linear.
Component: LineEquation
Props:
- title: Arc and Chord Relationship
- description: Comparison of arc with chord at various central angles.
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const angle = i * (Math.PI / 3) / 20;
return {
x: 5 * Math.cos(angle),
y: 5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("VIOLET"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Arc 60°", at: 10, offset: [1.5, 0.5, 0] }],
},
{
points: [
{ x: 5, y: 0, z: 0 },
{ x: 5 * Math.cos(Math.PI / 3), y: 5 * Math.sin(Math.PI / 3), z: 0 },
],
color: getColor("ORANGE"),
lineWidth: 3,
showPoints: false,
labels: [{ text: "Chord 60°", at: 0.5, offset: [0.8, -0.5, 0] }],
},
{
points: [
{ x: 5, y: 0, z: 0 },
],
color: getColor("PURPLE"),
showPoints: true,
labels: [{ text: "A", at: 0, offset: [0.5, -0.3, 0] }],
},
{
points: [
{ x: 5 * Math.cos(Math.PI / 3), y: 5 * Math.sin(Math.PI / 3), z: 0 },
],
color: getColor("PURPLE"),
showPoints: true,
labels: [{ text: "B", at: 0, offset: [0.3, 0.5, 0] }],
},
{
points: Array.from({ length: 31 }, (_, i) => {
const angle = Math.PI + (i * (2 * Math.PI / 3) / 30);
return {
x: 5 * Math.cos(angle),
y: 5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("ROSE"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Arc 120°", at: 15, offset: [-1.5, -0.5, 0] }],
},
{
points: [
{ x: -5, y: 0, z: 0 },
{ x: 5 * Math.cos ... [truncated; 2118 chars]
- cameraPosition: [0, 0, 14]
- showZAxis: false
## Chord Length Formula
The chord length can be calculated using a trigonometric formula involving the central angle and circle radius:
```math
c = 2r \sin\left(\frac{\theta}{2}\right)
```
Where:
- $$c$$ = chord length
- $$r$$ = circle radius
- $$\theta$$ = central angle in radians
Visible text: - = chord length
- = circle radius
- = central angle in radians
This formula is very useful in engineering calculations, especially in curved structure design and material strength analysis.
## Arc Height and Sagitta
Arc height or sagitta is the perpendicular distance from the chord midpoint to the highest point of the arc. This concept is very important in arch bridge design and architectural structures.
```math
h = r - r\cos\left(\frac{\theta}{2}\right) = r\left(1 - \cos\left(\frac{\theta}{2}\right)\right)
```
Where:
- $$h$$ = arc height (sagitta)
- $$r$$ = circle radius
- $$\theta$$ = central angle in radians
Visible text: - = arc height (sagitta)
- = circle radius
- = central angle in radians
Observe the following visualization:
Component: LineEquation
Props:
- title: Arc Height Visualization
- description: Arc height (sagitta) at various central angles.
- data: [
{
points: Array.from({ length: 31 }, (_, i) => {
const angle = (-Math.PI / 6) + (i * (Math.PI / 3) / 30);
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("VIOLET"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Arc 60°", at: 15, offset: [1.5, 1, 0] }],
},
{
points: [
{ x: 4 * Math.cos(-Math.PI / 6), y: 4 * Math.sin(-Math.PI / 6), z: 0 },
{ x: 4 * Math.cos(Math.PI / 6), y: 4 * Math.sin(Math.PI / 6), z: 0 },
],
color: getColor("ORANGE"),
lineWidth: 3,
showPoints: false,
labels: [{ text: "Chord", at: 0.5, offset: [0, -0.8, 0] }],
},
{
points: [
{ x: (4 * Math.cos(-Math.PI / 6) + 4 * Math.cos(Math.PI / 6)) / 2, y: (4 * Math.sin(-Math.PI / 6) + 4 * Math.sin(Math.PI / 6)) / 2, z: 0 },
],
color: getColor("EMERALD"),
showPoints: true,
labels: [{ text: "M (Midpoint)", at: 0, offset: [1.2, -1.2, 0] }],
},
{
points: [
{ x: (4 * Math.cos(-Math.PI / 6) + 4 * Math.cos(Math.PI / 6)) / 2, y: (4 * Math.sin(-Math.PI / 6) + 4 * Math.sin(Math.PI / 6)) / 2, z: 0 },
{ x: 4 * Math.cos(0), y: 4 * Math.sin(0), z: 0 },
],
color: getColor("ROSE"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Arc Height (h)", offset: [0.8, -0.5, 0] }],
},
{
points: [
{ x: 0, ... [truncated; 2387 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
## Arc in Coordinate System
In the Cartesian coordinate system, an arc can be represented using parametric equations:
Component: MathContainer
Children:
```math
x = r\cos(t)
```
```math
y = r\sin(t)
```
Where $$t$$ is a parameter that varies from the initial angle to the final angle of the arc.
Visible text: Where is a parameter that varies from the initial angle to the final angle of the arc.
Component: LineEquation
Props:
- title: Arc in Cartesian Coordinates
- description: Arc representation using parametric equations.
- data: [
{
points: Array.from({ length: 41 }, (_, i) => {
const t = (Math.PI / 4) + (i * (Math.PI / 2) / 40);
return {
x: 3 * Math.cos(t),
y: 3 * Math.sin(t),
z: 0,
};
}),
color: getColor("VIOLET"),
lineWidth: 4,
showPoints: false,
labels: [{ text: "Parametric Arc", at: 20, offset: [1.5, 1, 0] }],
},
{
points: [
{ x: 3 * Math.cos(Math.PI / 4), y: 3 * Math.sin(Math.PI / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "t₁ = π/4 (45°)", at: 0, offset: [2, 0.5, 0] }],
},
{
points: [
{ x: 3 * Math.cos(3 * Math.PI / 4), y: 3 * Math.sin(3 * Math.PI / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "t₂ = 3π/4 (135°)", at: 0, offset: [-2, 0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3 * Math.cos(Math.PI / 4), y: 3 * Math.sin(Math.PI / 4), z: 0 },
],
color: getColor("PURPLE"),
lineWidth: 2,
showPoints: false,
labels: [{ text: "r = 3", at: 0.5, offset: [0.3, -0.3, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3 * Math.cos(3 * Math.PI / 4), y: 3 * Math.sin(3 * Math.PI / 4), z: 0 },
],
color: getColor("PURPLE"),
lineWidth: 2,
showPoints: false,
labels: [{ text: "r = 3", at: 0.5, offset: [-0.3, -0.3, 0] }],
},
{
points: [
{ x: 0, y: ... [truncated; 1566 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
## Engineering Calculation Example
Let's apply this concept in engineering calculations. Suppose we design an arch bridge with a radius of $$25 \text{ meters}$$ and a central angle of $$120^\circ$$.
Visible text: Let's apply this concept in engineering calculations. Suppose we design an arch bridge with a radius of and a central angle of .
**Calculating chord length:**
Component: MathContainer
Children:
```math
c = 2r \sin\left(\frac{\theta}{2}\right)
```
```math
c = 2 \times 25 \times \sin\left(\frac{120^\circ}{2}\right)
```
```math
c = 50 \times \sin(60^\circ)
```
```math
c = 50 \times \frac{\sqrt{3}}{2} = 25\sqrt{3} \text{ meters}
```
**Calculating arc height:**
Component: MathContainer
Children:
```math
h = r\left(1 - \cos\left(\frac{\theta}{2}\right)\right)
```
```math
h = 25\left(1 - \cos(60^\circ)\right)
```
```math
h = 25\left(1 - \frac{1}{2}\right) = 12.5 \text{ meters}
```
## Exercises
1. An arch bridge has a radius of $$30 \text{ meters}$$ and a central angle of $$90^\circ$$. Calculate the chord length and arc height of the bridge.
2. In a mosque dome design, the arc height is $$8 \text{ meters}$$ and the circle radius is $$15 \text{ meters}$$. Determine the central angle of the arc.
3. A circle arc has a chord length of $$24 \text{ meters}$$ and a radius of $$15 \text{ meters}$$. Calculate the central angle and arc height.
4. In a coordinate system, an arc starts from point $$(4, 0)$$ and ends at point $$(0, 4)$$ on a circle centered at the origin. Determine the parametric equations of the arc.
Visible text: 1. An arch bridge has a radius of and a central angle of . Calculate the chord length and arc height of the bridge.
2. In a mosque dome design, the arc height is and the circle radius is . Determine the central angle of the arc.
3. A circle arc has a chord length of and a radius of . Calculate the central angle and arc height.
4. In a coordinate system, an arc starts from point and ends at point on a circle centered at the origin. Determine the parametric equations of the arc.
### Answer Key
1. **Solution:**
Given: $$r = 30 \text{ m}$$ and $$\theta = 90^\circ$$
**Chord length:**
```math
c = 2r \sin\left(\frac{\theta}{2}\right) = 2 \times 30 \times \sin(45^\circ)
```
```math
c = 60 \times \frac{\sqrt{2}}{2} = 30\sqrt{2} \text{ m} \approx 42.43 \text{ m}
```
**Arc height:**
```math
h = r\left(1 - \cos\left(\frac{\theta}{2}\right)\right) = 30\left(1 - \cos(45^\circ)\right)
```
```math
h = 30\left(1 - \frac{\sqrt{2}}{2}\right) \approx 8.79 \text{ m}
```
2. **Solution:**
Given: $$h = 8 \text{ m}$$ and $$r = 15 \text{ m}$$
**Step** $$1$$: Use the arc height formula
```math
8 = 15\left(1 - \cos\left(\frac{\theta}{2}\right)\right)
```
**Step** $$2$$: Isolate $$\cos$$
```math
\frac{8}{15} = 1 - \cos\left(\frac{\theta}{2}\right)
```
```math
\cos\left(\frac{\theta}{2}\right) = 1 - \frac{8}{15} = \frac{7}{15}
```
**Step** $$3$$: Calculate angle
```math
\frac{\theta}{2} = \arccos\left(\frac{7}{15}\right) \approx 62.18^\circ
```
```math
\theta \approx 124.36^\circ
```
3. **Solution:**
Given: $$c = 24 \text{ m}$$ and $$r = 15 \text{ m}$$
**Step** $$1$$: Use the chord formula
```math
24 = 2 \times 15 \times \sin\left(\frac{\theta}{2}\right)
```
```math
\sin\left(\frac{\theta}{2}\right) = \frac{24}{30} = 0.8
```
**Step** $$2$$: Calculate central angle
```math
\frac{\theta}{2} = \arcsin(0.8) \approx 53.13^\circ
```
```math
\theta \approx 106.26^\circ
```
**Step** $$3$$: Calculate arc height
```math
h = 15\left(1 - \cos(53.13^\circ)\right) = 15(1 - 0.6) = 6 \text{ m}
```
4. **Solution:**
- Starting point: $$(4, 0)$$ → $$t_1 = 0^\circ$$
- Ending point: $$(0, 4)$$ → $$t_2 = 90^\circ$$
- Radius: $$r = 4$$
**Parametric equations:**
```math
x = 4\cos(t)
```
```math
y = 4\sin(t)
```
```math
0^\circ \leq t \leq 90^\circ
```
Visible text: 1. **Solution:**
Given: and
**Chord length:**
**Arc height:**
2. **Solution:**
Given: and
**Step** : Use the arc height formula
**Step** : Isolate
**Step** : Calculate angle
3. **Solution:**
Given: and
**Step** : Use the chord formula
**Step** : Calculate central angle
**Step** : Calculate arc height
4. **Solution:**
- Starting point: →
- Ending point: →
- Radius:
**Parametric equations:**