Central Angle
In short: A central angle is an opening pinned at the center of a circle, formed by two radii, and it measures the same number of degrees as the stretch of rim it spans.
GeometryGeometryG-C.A.2
What is Central Angle?
Plant the point of a compass at the middle of a circle and spread two radii apart: the gap between them is a central angle. Its corner sits at the center, and both of its arms end on the rim.
The key rule is a one-to-one match with the rim stretch it spans: the degrees are equal. A \(108^{\circ}\) opening spans \(108^{\circ}\) of rim and a fraction \(\frac{108}{360} = \frac{3}{10}\) of the whole circle. That fraction also tells how much of the circumference and how much of the area the slice holds.
How to spot it
- A figure shows two radii from the center of a circle with an angle mark between them.
- A pie chart or spinner is divided into wedges and each wedge needs its opening in degrees.
- The words sector, wedge or slice appear with a percentage or fraction of a whole.
- An arc measure is given and the matching opening at the center is requested.
The method
- Confirm the vertex is at the center, so the sides are radii.
- Write the part as a fraction of the whole: part divided by total, or the stated percentage.
- Multiply that fraction by \(360^{\circ}\) to get the opening.
- Set the arc measure equal to the opening, and use it for arc length or sector area if needed.
Central Angle: solved example
Problem
A family’s monthly budget of $4,200 is shown in a pie chart. Rent takes $1,260. What central angle should the rent wedge have?
Solution
The rent share is \(\frac{1{,}260}{4{,}200} = 0.3\).
Multiply by the full turn: \(0.3 \times 360 = 108\). The rent wedge needs a central angle of \(108^{\circ}\), and its arc measures \(108^{\circ}\) too.
Try it
Question 1
A central angle measures \(135^{\circ}\). What fraction of the circle is the matching slice?
Show the answer
\(\frac{135}{360} = \frac{3}{8}\) of the circle.
Question 2
A spinner wedge covers 15% of the circle. How many degrees is its central angle?
Show the answer
\(0.15 \times 360 = 54\), so the angle is \(54^{\circ}\).
Central Angle vs. Inscribed Angle
The corner’s location marks the difference. A central angle is pinned at the center and equals its arc. An inscribed angle is pinned on the rim and equals only half of the arc it cuts off. For a \(108^{\circ}\) arc, the first reads \(108^{\circ}\) and the second reads \(54^{\circ}\).
Keep going
Lessons
Worksheets and quizzes
- Circles: Angles, Chords and Tangents – Worksheet
- Circles: Angles, Chords and Tangents – Quiz
- Arc Length and Sector Area – Worksheet
- Arc Length and Sector Area – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry