Key takeaway: A central angle is an angle whose vertex is the center of a circle and whose sides are radii, and its measure equals the measure of the arc it intercepts.
Grade 10 – GeometryG-C.A.2Geometry and Measurement
Definition of Central Angle
A central angle has its corner at the middle of a circle, with both sides running out to the circle as radii. The part of the circle between the two sides is the intercepted arc.
The rule is direct: the angle and its arc have the same degree measure. A quarter turn of \(90^{\circ}\) cuts off a quarter of the circle, and a half turn of \(180^{\circ}\) cuts off a half. Pie charts rely on this idea: each slice’s angle equals its share of 360 degrees.
Formula and notation
If \(O\) is the center and \(A\), \(B\) lie on the circle:
\[ m\angle AOB = m\overset{\frown}{AB} \]
For a share \(p\) of the whole circle, written as a decimal, the angle is \(p \times 360^{\circ}\).
Key facts
- The vertex must sit exactly at the center of the circle.
- Its measure equals its intercepted arc, unlike an inscribed angle, which is half.
- All the angles around the center add up to 360 degrees.
- Equal angles at the center cut off equal arcs and equal chords.
Central Angle example
In a lunch survey, 18% of students chose tacos. In a pie chart, what angle at the center should the taco slice have? Also, how large is the arc if another slice has an angle of \(117^{\circ}\)?
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Share as a decimal: \(18\% = 0.18\).
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Taco slice: \(0.18 \times 360^{\circ} = 64.8^{\circ}\).
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The arc of the other slice has the same measure as its central angle, \(117^{\circ}\); that slice is \(\frac{117}{360} = 32.5\%\) of the pie.
The taco slice needs \(64.8^{\circ}\) at the center; the \(117^{\circ}\) slice has a \(117^{\circ}\) arc and shows 32.5%.
Common mistake
Watch out
Cleo draws the taco slice with a \(18^{\circ}\) angle, since the survey says 18%.
Why it is wrong
A percent of the pie is a fraction of the full turn, \(360^{\circ}\), not of \(100^{\circ}\). Multiply \(0.18 \times 360 = 64.8^{\circ}\). A check: 100% of the survey must add to a full \(360^{\circ}\), not \(100^{\circ}\).
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement