What Is the Central Angle of a Circle?
A central angle is an angle whose vertex sits at the center of a circle and whose two sides are radii reaching out to the circle's edge. Written $\angle AOB$, with $O$ the center, it opens onto the arc $AB$ trapped between the two radii.
The central angle and its arc are two readings of the same thing: a bigger angle sweeps a longer arc, in exact proportion. A full turn around the center is $360°$ (or $2\pi$ radians) and it opens onto the whole circumference. This article focuses on the formula linking the angle to its arc; for the broader idea and its angle relationships, see central angle in geometry.
The Angle a Clock's Hands Sweep Without Anyone Noticing
Every minute, the minute hand of a clock swings through exactly $6°$ at the center of the dial, and every hour the hour hand creeps through $30°$. Those are central angles, the angle measured right at the pivot in the middle. The same idea sizes a slice of pizza, the reach of a radar sweep, and the bend of a highway curve. Get the one formula that ties a central angle to the arc it cuts, and all of them become the same calculation.
What Is the Central Angle Formula?
There are two forms, one for each way of measuring angles. Define the variables first: $\theta$ is the central angle, $s$ the arc length, $r$ the radius, and $\pi \approx 3.14159$.
In radians, the clean form:
$$\theta = \frac{s}{r}$$
This is where radians come from. One radian is the angle whose arc equals the radius, so dividing the arc by the radius counts "how many radius-lengths of arc", that count is the angle in radians. Rearranged, it gives the arc-length formula:
$$s = r\theta \quad (\theta \text{ in radians})$$
In degrees, the arc as a fraction of the whole circle:
$$\theta = \frac{s}{2\pi r} \times 360°$$
The arc $s$ is some fraction of the full circumference $2\pi r$, and that same fraction of $360°$ is the central angle. If you already know the arc length and the circumference, this is just "what share of the circle is this arc, in degrees."
To move between the two, convert: radians $= \text{degrees} \times \frac{\pi}{180}$, and degrees $= \text{radians} \times \frac{180}{\pi}$.
How Do You Find the Central Angle?
Pick the form that matches what you are given.
Given arc length and radius, want radians: divide, $\theta = \frac{s}{r}$.
Given arc length and radius, want degrees: use $\theta = \frac{s}{2\pi r} \times 360°$.
Given the arc as a fraction of the circle: multiply that fraction by $360°$. A quarter-circle arc gives $\frac{1}{4} \times 360° = 90°$.
Given a sector's share of the area: the central angle is the same fraction of $360°$ as the sector is of the whole circle.
What Is the Central Angle Theorem?
The central angle also connects to angles drawn from the edge of the circle. The central angle theorem says a central angle is exactly twice any inscribed angle that opens onto the same arc:
$$\theta_{\text{central}} = 2 \times \theta_{\text{inscribed}}$$
An inscribed angle has its vertex on the circle rather than at the center. So if an inscribed angle over some arc measures $35°$, the central angle over that same arc measures $70°$. This halving-and-doubling relationship is one of the most-used facts in circle geometry, and it is why a diameter always subtends a right angle from the circle's edge (a $180°$ central angle gives a $90°$ inscribed angle).
What Are the Properties of the Central Angle?
The central angle carries a set of properties worth holding together.
Proportional to its arc. Double the arc and you double the central angle, arc and angle rise together in lockstep.
A full revolution is $360°$ (or $2\pi$ radians), opening onto the entire circumference.
Twice any inscribed angle on the same arc, by the central angle theorem.
Equal chords subtend equal central angles, and equal central angles cut off equal arcs.
Sets the sector's share. A sector's area is $\frac{\theta}{360°} \times \pi r^2$, the same fraction of the circle that the angle is of $360°$.
Examples of Central Angle of a Circle Formula
Example 1
A circle has radius $r = 5$ and an arc of length $s = 10$. Find the central angle in radians.
Divide the arc by the radius:
$$\theta = \frac{s}{r} = \frac{10}{5} = 2 \text{ radians}$$
Final answer: the central angle is $2$ radians.
Example 2
A radius-$6$ circle has an arc that subtends a central angle of $90°$. Find the arc length.
Wrong attempt. Reach straight for $s = r\theta$: $s = 6 \times 90 = 540$. But the whole circumference is only $2\pi \times 6 \approx 37.7$, so an arc of $540$ is impossible, it is more than fourteen times around the circle.
Break. The formula $s = r\theta$ needs $\theta$ in radians, not degrees. Plugging in $90$ (degrees) breaks it.
Correct. Convert first: $90° = 90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians. Then:
$$s = r\theta = 6 \times \frac{\pi}{2} = 3\pi \approx 9.42$$
Or use the degree form directly: $s = \frac{90°}{360°} \times 2\pi \times 6 = \frac{1}{4} \times 37.7 \approx 9.42$. Both agree.
Final answer: the arc length is $3\pi \approx 9.42$ units.
Example 3
An arc of length $s = 14$ lies on a circle of radius $r = 7$. Find the central angle in degrees.
Use the degree formula:
$$\theta = \frac{s}{2\pi r} \times 360° = \frac{14}{2\pi \times 7} \times 360° = \frac{14}{14\pi} \times 360° = \frac{360°}{\pi} \approx 114.6°$$
Final answer: the central angle is about $114.6°$.
Example 4
An inscribed angle opens onto an arc and measures $40°$. What is the central angle over the same arc?
By the central angle theorem, the central angle is twice the inscribed angle:
$$\theta_{\text{central}} = 2 \times 40° = 80°$$
Final answer: the central angle is $80°$.
Example 5
A pizza is cut into $8$ equal slices. What central angle does each slice make?
Each slice is $\frac{1}{8}$ of the full $360°$ turn:
$$\theta = \frac{360°}{8} = 45°$$
Final answer: each slice has a central angle of $45°$.
Example 6
A sector of a circle covers $\frac{1}{6}$ of the circle's area. Find its central angle in degrees.
The central angle is the same fraction of $360°$ as the sector is of the whole circle:
$$\theta = \frac{1}{6} \times 360° = 60°$$
Final answer: the central angle is $60°$.
Where Is the Central Angle Used?
The central angle is how curved space gets divided fairly and measured precisely.
Pie and donut charts. Each category's slice gets a central angle equal to its share of the whole, out of $360°$.
Clocks and gears. Hands and cog teeth are positioned by equal central angles around a hub.
Road and railway design. A curved section of road is specified by its radius and central angle, which together fix the arc length drivers actually travel.
Radar and astronomy. Angular sweeps and the apparent positions of objects are measured as central angles from a fixed observation point. The formal reference for the definition is catalogued at Wolfram MathWorld.
The reason the same formula fits all of them is that a central angle is nothing more than "what fraction of the full turn is this," measured at the center.
Where Intuition Breaks on the Central Angle
Mistake 1: Using s = rθ with the angle in degrees
Where it slips in: finding arc length when the central angle is given in degrees.
Don't do this: plugging a degree value straight into $s = r\theta$.
The correct way: $s = r\theta$ is a radian-only formula. Convert degrees to radians first ($\times \frac{\pi}{180}$), or switch to the degree form $s = \frac{\theta}{360°} \times 2\pi r$. The single most common error on this topic is dropping a degree measure into the radian formula, always confirm which unit your angle is in before you compute.
Mistake 2: Confusing the central angle with the inscribed angle
Where it slips in: angle problems that mix a vertex at the center with a vertex on the circle.
Don't do this: treating an inscribed angle as equal to the central angle over the same arc.
The correct way: the central angle is twice the inscribed angle on the same arc, not equal to it. The memorizer who recalls "angle over the arc" but not which vertex it uses drops the factor of two. Check where the vertex sits, center or edge, before applying anything.
Mistake 3: Forgetting the central angle caps at 360°
Where it slips in: working backward from a very long arc.
Don't do this: reporting a central angle larger than $360°$ (or $2\pi$ radians) for a single sweep.
The correct way: a central angle for one revolution never exceeds $360°$. If a calculation gives more, the arc length or radius was mis-entered, a full circle is the ceiling. This is a fast reasonableness check the rusher skips.
Conclusion
The central angle of a circle formula is $\theta = \frac{s}{r}$ in radians and $\theta = \frac{s}{2\pi r} \times 360°$ in degrees.
Rearranged, arc length is $s = r\theta$, but only when $\theta$ is measured in radians.
The central angle is twice any inscribed angle on the same arc (the central angle theorem).
A central angle is the same fraction of $360°$ that its arc is of the circumference, or its sector is of the area.
The classic error is dropping a degree measure into $s = r\theta$; convert to radians first.
To take the central angle of a circle formula further with a teacher, explore Bhanzu's geometry tutor, a high school math tutor, or live math classes online.
Practice These to Solidify Your Understanding
Work through these three, checking each against the circumference: (1) find the central angle in radians for an arc of length $12$ on a radius-$4$ circle; (2) find the arc length for a $120°$ central angle on a radius-$9$ circle; (3) find the central angle over the same arc as a $25°$ inscribed angle. If a degree-radian mix trips you, return to What Is the Central Angle Formula? above. Want a live Bhanzu trainer to walk through more arc-and-angle problems? Book a free demo class.
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