Circle Measurements and Graphs: Arcs

In this lesson, we will explore the concept of arcs in circles. An arc is a portion of the circumference of a circle. The measure of an arc is determined by the angle it subtends at the center of the circle, measured in degrees or radians.

To find the length of an arc, we can use the formula:

$$L = \frac{\theta}{360} \times C$$

where:

• $L$ is the length of the arc,
• $\theta$ is the central angle in degrees,
• $C$ is the circumference of the circle.

The circumference of a circle can be calculated using:

$$C = 2\pi r$$

where $r$ is the radius of the circle.

Example

Let's find the length of an arc with a central angle of $60^\circ$ in a circle with a radius of 4 units.

1. Calculate the circumference of the circle: $$C = 2\pi r = 2\pi \times 4 = 8\pi$$
2. Use the arc length formula: $$L = \frac{60}{360} \times C = \frac{60}{360} \times 8\pi$$
3. Simplify the fraction: $$L = \frac{1}{6} \times 8\pi = \frac{8\pi}{6} = \frac{4\pi}{3}$$

Thus, the length of the arc is $\frac{4\pi}{3}$ units.

Key Questions

• What is the formula for the circumference of a circle?
• How do you calculate the length of an arc given the central angle and radius?
• If the radius of a circle is doubled, how does that affect the length of the arc?
• What is the relationship between the central angle and the length of the arc?

Keywords

• Arc
• Circumference
• Central angle
• Length of an arc
• Circle