Find arc lengths from central angles and radius on a circle

Key takeaway: Arc-length questions are really about one formula, but the trick is noticing which angle belongs to which arc. Once the central angle is known in radians, the rest is mechanical.

For a circle, arc length is

$$s=r\theta$$

where:

• $s$ = arc length
• $r$ = radius
• $\theta$ = central angle in radians

From the prompt, the circle is centered at $O$ and the radius is the segment labeled to the circle. The instruction says to find each arc length to the nearest hundredth.

How to solve

1. Find the central angle for the arc.
2. Multiply by the radius.
3. Round to the nearest hundredth.

If the angle is already given in radians, the computation is direct.

For example, if an arc has radius $r=10.3$ and central angle $\theta$, then

$$s=10.3\theta$$

So the entire task reduces to matching each arc with its corresponding central angle.

Final form

• Arc $CD$: $s=13\cdot \theta_{CD}$ if the radius is $13$ cm.
• Arc $AB$: $s=97\cdot \theta_{AB}$ if the radius is $97$ ft.

If the diagram provides the angles, substitute them directly into $s=r\theta$.

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Pitfalls the pros know 👇 A frequent mistake is using degrees without converting to radians. Another one is mixing up radius and diameter. Arc length uses the radius, not the full width of the circle.

What if the problem changes? If the problem gives a central angle in degrees, convert first:

$$\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$$

Then use $s=r\theta_{rad}$. If the arc is a major arc, remember the angle may be $2\pi$ minus the smaller central angle.

Tags: arc length, radians, central angle