To find the central angle of a circle given the radius and arc length, you use a simple formula that relates these three quantities. The central angle is directly proportional to the arc length it subtends and inversely proportional to the radius of the circle.
The most common formula for finding the central angle ($\theta$) when the arc length ($s$) and radius ($r$) are known is:
$$ \theta = \frac{s}{r} $$
In this formula, the angle $\theta$ is always measured in radians.
Understanding the Formula
This relationship comes from the definition of a radian. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius.
- $\theta$: The central angle (in radians).
- $s$: The arc length (the length of the curved line along the edge of the circle).
- $r$: The radius of the circle (the distance from the center to any point on the edge).
Key Point: Ensure that the arc length ($s$) and radius ($r$) are in the same units (e.g., both in centimeters, both in inches). The resulting angle $\theta$ will be in radians. If you need the angle in degrees, you can convert radians to degrees using the conversion factor: 1 radian $\approx 57.3$ degrees, or simply multiply the radian measure by $\frac{180}{\pi}$.
Applying the Formula: An Example
Let's look at an example, similar to what is shown in the reference video:
Suppose you are given:
- Arc length ($s$) = 8 units
- Radius ($r$) = 5 centimeters
And you need to find the central angle ($\theta$).
Using the formula:
$$ \theta = \frac{s}{r} $$
Substitute the given values:
$$ \theta = \frac{8}{5} $$
$$ \theta = 1.6 \text{ radians} $$
So, the central angle is 1.6 radians.
Steps to Calculate the Central Angle
Here's a simple breakdown of the steps:
- Identify the given values: Determine the arc length ($s$) and the radius ($r$).
- Ensure consistent units: Make sure $s$ and $r$ are measured using the same units (e.g., meters, feet, cm).
- Use the formula: Divide the arc length by the radius ($\theta = s / r$).
- State the units: The resulting angle is in radians.
Summary Table
| Component | Symbol | Units | Description |
|---|---|---|---|
| Central Angle | $\theta$ | Radians | The angle at the center of the circle. |
| Arc Length | $s$ | e.g., cm, m | The length of the curved part of the circle. |
| Radius | $r$ | e.g., cm, m | The distance from the center to the circumference. |
This straightforward formula allows you to easily determine the central angle once you know the arc length and the circle's radius.
