What if the angle is given as a major arc (greater than 180 degrees)?

The formulas remain identical. Use the given major angle for $ heta$ to find the major arc length and major sector area.

How do I handle negative angles?

Arc length and area are scalar quantities and strictly positive. Take the absolute value of the angle before applying the formulas.

What if I am given the arc length and need to find the sector area?

Use the relation $A = \frac{1}{2}rs$. Substitute the known radius and arc length directly to solve for area without needing to find the central angle first.

How to find arc length and sector area

To find the arc length and sector area of a circle, scale the circumference and total area by the fraction of the central angle relative to a full circle.

This method applies whenever the radius of the circle and the central angle defining the sector are known.

The setup

Identify the radius $r$ of the circle and the central angle $heta$ subtending the arc. Note whether $heta$ is measured in degrees or radians.

The steps

If $heta$ is in degrees, calculate arc length as $s = 2\pi r (\frac{ heta}{360})$ and sector area as $A = \pi r^2 (\frac{ heta}{360})$ .
If $heta$ is in radians, calculate arc length as $s = r heta$ and sector area as $A = \frac{1}{2}r^2 heta$ .
Simplify the resulting exact expressions, keeping $\pi$ unless a decimal approximation is specified.

Checking the result

Verify that the ratio of your calculated arc length to $2\pi r$ exactly equals the ratio of your calculated sector area to $\pi r^2$ . Both ratios must equal the fraction of the circle subtended by $heta$ .

Common errors

The most frequent error is applying the radian formulas $s = r heta$ or $A = \frac{1}{2}r^2 heta$ when $heta$ is given in degrees. Another common mistake is using the diameter instead of the radius.

Worked example

Find the exact arc length and sector area of a circle with radius $r = 6$ cm and central angle $heta = 120^\circ$ .

Given: $r = 6$ , $heta = 120^\circ$ .

Step 1: The angle is in degrees.

Step 2: Calculate arc length. $s = 2\pi r (\frac{ heta}{360})$ $s = 2\pi (6) (\frac{120}{360})$ $s = 12\pi (\frac{1}{3})$ $s = 4\pi$ cm.

Step 3: Calculate sector area. $A = \pi r^2 (\frac{ heta}{360})$ $A = \pi (6^2) (\frac{120}{360})$ $A = 36\pi (\frac{1}{3})$ $A = 12\pi$ cm $^2$ .

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References: OpenStax Precalculus, Chapter 5 · Khan Academy: High School Geometry - Circles