An angle whose vertex is at the center of a circle, with its two rays intersecting the circle at two points. Its measure equals the measure of the intercepted arc.
Imagine standing at the center of a clock face. The angle between the hour and minute hands is a central angle. The arc between the two numbers the hands point to is the intercepted arc, and its measure (in degrees) equals the angle you see.
Showing a random 20 of 50 problems.
Example 1
easy
A pie chart slice represents 50% of the data. What is its central angle?
Example 2
challenge
A regular hexagon is inscribed in circle O of radius r. What is the central angle that one side of the hexagon subtends?
Example 3
medium
A circle is divided into central angles in the ratio 2:3:4. Find each angle.Ratio 2:3:4 → sectors 80°, 120°, 160° (sum 360°)
Example 4
medium
An arc is 1/8 of a circle. What is its central angle?1/8 of circle → x = 360°/8 = 45°
Example 5
easy
The central angles around a circle's center add up to what?
Example 6
easy
Three central angles in a circle measure 90°, 130°, and x. They make up the whole circle. Find x.
Example 7
challenge
A chord of length 12 subtends a central angle in a circle of radius 10. Find the central angle (to the nearest degree).Chord=12, radius=10 → central angle x ≈ 74°
Example 8
easy
An arc measures 130°. What is the measure of the central angle that intercepts it?Arc = 130° → central angle x = 130°
Example 9
easy
A pie chart shows a category that is 25% of the data. What central angle represents it?25% of circle → central angle x = 90°
Example 10
easy
Where is the vertex of a central angle?
Example 11
medium
Convert a central angle of 60° into radians.
Example 12
easy
On an analog clock, the minute hand moves from 12 to 3. Through what central angle does it sweep?
Example 13
easy
A central angle measures 45°. What is the measure of the arc it intercepts?
Example 14
medium
At what time after 12:00 do the clock hands form a 90° central angle for the first time? (Approximate.)
Example 15
hard
Two central angles in a circle are supplementary (90° and 90° would be a degenerate case). If ∠AOB+∠COD=180° and ∠AOB=70°, find ∠COD.
Example 16
medium
In circle O, central angles ∠AOB and ∠BOC measure 50° and 70°. Find ∠AOC, assuming B is between A and C on the same arc.
Example 17
hard
At 3:30 on an analog clock, what is the angle between the hour hand and the minute hand?
Example 18
medium
A regular hexagon is inscribed in a circle. What central angle subtends each side?Regular hexagon: 360° ÷ 6 = 60° central angle per side
Example 19
medium
A sector has central angle 72°. What fraction of the circle does it represent?
Example 20
medium
A pie chart shows budget categories. Housing is 40%, Food is 25%, Transport is 15%, Other is the rest. Find the central angle for Other.