Circle Arc

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.

For minor arcs in the same circle, a larger central angle produces both a longer arc and a longer chord, but the relationship is not linear. Beyond $180^\circ$, the major arc keeps getting longer while its chord becomes shorter, because the chord is shared with the complementary minor arc.

Chord Length Formula

The radius and central angle determine the chord length:

The variables in this formula are:

• $c$ = chord length
• $r$ = circle radius
• $\theta$ = central angle. Use degree mode for degrees and radian mode for radians

The formula determines a chord from a circular arc's radius and central angle. It can specify the geometry of an idealized circular window, arch, or other curved outline. A real structure still requires a separate analysis of its loads, materials, and supports.

Arc Height and Sagitta

Arc height, or sagitta, is the distance from the chord's midpoint to the corresponding minor arc, measured perpendicular to the chord. Together with the chord, it specifies the shape of an idealized circular arc.

The variables in this formula are:

• $h$ = arc height (sagitta)
• $r$ = circle radius
• $\theta$ = central angle. Evaluate the cosine in the same angle unit used for $\theta$

The diagram marks the chord, its midpoint, and the perpendicular segment whose length is the sagitta.

Arc in Coordinate System

In the Cartesian coordinate system, an arc can be represented using parametric equations:

Where $t$ is a parameter that varies from the initial angle to the final angle of the arc.

Idealized Circular Arch Example

An idealized circular arch has radius $25 \text{ meters}$ and central angle $120^\circ$. The chord gives its span, and the sagitta gives its rise.

Calculating chord length:

Calculating arc height:

Exercises

Each problem mixes a circle measurement with an arc measurement, so mark which quantity is given before you choose the formula.

1. An idealized circular arch has a radius of $30 \text{ m}$ and a central angle of $90^\circ$. Calculate its chord length and arc height.

2. In an idealized circular cross-section of a dome, the arc height is $8 \text{ m}$ and the circle radius is $15 \text{ m}$. Determine the central angle of the arc.

3. A minor circular arc has a chord length of $24 \text{ m}$ and a radius of $15 \text{ m}$. Calculate its minor 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 a standard radian parameterization of the minor arc.

Worked Solutions

1. Solution:

   Given: $r = 30 \text{ m}$ and $\theta = 90^\circ$

   Chord length:

   $$\begin{aligned} c &= 2r \sin\left(\frac{\theta}{2}\right) \\ &= 2 \times 30 \times \sin(45^\circ) \end{aligned}$$

   $$c = 60 \times \frac{\sqrt{2}}{2} = 30\sqrt{2} \text{ m} \approx 42.43 \text{ m}$$

   Arc height:

   $$\begin{aligned} h &= r\left(1 - \cos\left(\frac{\theta}{2}\right)\right) \\ &= 30\left(1 - \cos(45^\circ)\right) \end{aligned}$$

   $$h = 30\left(1 - \frac{\sqrt{2}}{2}\right) \approx 8.79 \text{ m}$$

2. Solution:

   Given: $h = 8 \text{ m}$ and $r = 15 \text{ m}$

   Step 1: Use the arc height formula

   $$8 = 15\left(1 - \cos\left(\frac{\theta}{2}\right)\right)$$

   Step 2: Isolate $\cos$

   $$\frac{8}{15} = 1 - \cos\left(\frac{\theta}{2}\right)$$

   $$\cos\left(\frac{\theta}{2}\right) = 1 - \frac{8}{15} = \frac{7}{15}$$

   Step 3: Calculate angle

   $$\frac{\theta}{2} = \arccos\left(\frac{7}{15}\right) \approx 62.18^\circ$$

   $$\theta \approx 124.36^\circ$$

3. Solution:

   Given: $c = 24 \text{ m}$ and $r = 15 \text{ m}$

   Step 1: Use the chord formula

   $$24 = 2 \times 15 \times \sin\left(\frac{\theta}{2}\right)$$

   $$\sin\left(\frac{\theta}{2}\right) = \frac{24}{30} = 0.8$$

   Step 2: Calculate the minor central angle

   $$\frac{\theta}{2} = \arcsin(0.8) \approx 53.13^\circ$$

   $$\theta \approx 106.26^\circ$$

   Step 3: Calculate arc height

   $$\begin{aligned} h &= 15\left(1 - \cos(53.13^\circ)\right) \\ &= 15(1 - 0.6) \\ &= 6 \text{ m} \end{aligned}$$

4. Solution:

   • Starting point: $(4, 0)$ → $t_1 = 0$
   • Ending point: $(0, 4)$ → $t_2 = \frac{\pi}{2}$
   • Radius: $r = 4$

   Parametric equations:

   $$x = 4\cos(t)$$

   $$y = 4\sin(t)$$

   $$0 \leq t \leq \frac{\pi}{2}$$