Central Angle: Definition, Circle Examples, Chart, and Practice

What is a central angle?

A central angle is an angle with its vertex at the center of a circle.

The two sides of the angle are radii because each side goes from the center to the circle.

The angle opens toward part of the circle called an arc.

Central angle rule

For a central angle, the angle measure matches the measure of the arc it opens toward.

If a central angle is 80°, then its intercepted arc is also 80°.

This works because the angle starts exactly at the center, so it controls that part of the circle directly.

Central angle chart

The chart shows the parts you should name in a central-angle diagram.

The radius points from the center to the circle, the arc sits on the circle, and the sector is the filled slice inside the two radii.

Central angle vs other circle lines

A central angle uses two radii and stays inside the circle.

A circle tangent is different because it touches the circle from outside at one point.

If the vertex is not at the center, the angle is not a central angle.

Worked example: find the arc measure

Problem: A central angle measures 115°. What is the measure of the arc it opens toward?

Step 1: Check that the angle starts at the center of the circle.

Step 2: Use the central angle rule: central angle measure = intercepted arc measure.

Answer: The arc measure is 115°.

Using a central angle for arc length

Arc measure tells how much of the circle is used. Arc length tells the actual distance along the curve.

Arc length = central angle / 360 × circumference

Since circumference is 2πr, you can write arc length = central angle / 360 × 2πr.

Example: arc length

Problem: A circle has radius 6 cm. A central angle is 90°. Find the arc length.

Step 1: A 90° angle is one fourth of a full 360° circle.

Step 2: Circumference = 2πr = 2π(6) = 12π.

Step 3: Arc length = 90 / 360 × 12π = 3π.

Answer: The arc length is 3π cm, or about 9.42 cm.

Sector area from a central angle

A sector is the slice of the circle inside the central angle.

Sector area = central angle / 360 × πr²

If the central angle is half the circle, the sector area is half the area of the circle. If it is one fourth of the circle, the sector area is one fourth of the area.

Common mistakes

Do not call an angle central unless its vertex is at the center of the circle.

Do not mix up arc measure and arc length. Arc measure is in degrees. Arc length is a distance.

Do not forget to divide by 360 when using arc length or sector area formulas.

Quick practice

1. A 70° central angle intercepts a 70° arc.

2. A 180° central angle makes a semicircle.

3. A 60° central angle uses one sixth of the circle.

4. If the vertex is on the circle instead of at the center, the angle is not central.