How to find a circle’s radius and sector area from arc length

Key concept: When arc length is given, the trick is to work backward through the arc formula first. After that, the sector area usually falls into place with the same angle.

Step by step

We are told the arc length is $32.5$ cm. That means we can use the arc-length formula to find the radius first.

Step 1: Use the arc-length formula

$s=\frac{\theta}{360^\circ}\cdot 2\pi r$

Here:

• $s=32.5$ cm
• $\theta$ is the central angle shown in the diagram
• $r$ is the unknown radius

Substitute the values from the circle shown and solve for $r$.

Step 2: Round the radius

Round your radius to the nearest tenth, as requested.

Step 3: Find the area of the shaded sector

Use the sector-area formula:

$A=\frac{\theta}{360^\circ}\cdot \pi r^2$

Since the same central angle is used, you can reuse the angle from Step 1 and the radius you found.

Step 4: Round the area

Round the shaded sector area to the nearest tenth.

A good final solution should clearly show:

• the equation used for arc length
• the radius calculation
• the sector-area calculation
• the rounded answers

If the diagram’s angle is given in radians, switch to $s=r\theta$ and $A=\frac12 r^2\theta$ instead.

Pitfall alert

Students often try to find the sector area directly from the arc length without first recovering the radius. That can work only if you set up the equations carefully. Another frequent error is rounding too early; keep extra digits until the final step.

Try different conditions

If the arc length changes but the central angle stays the same, the radius changes too. If the question instead gives the radius and arc length, you can solve for the central angle first, then use that same angle for the sector area. The structure is the same; only the unknown changes.

Further reading

sector area, arc length, central angle