What Is a Central Angle in Geometry?
A central angle in geometry is an angle whose vertex is the centre of a circle and whose two sides are radii that meet the circle at two points. The arc lying between those two points, the part of the rim the angle "opens onto," is called the intercepted arc.
The defining property is simple and exact: the measure of a central angle equals the measure of its intercepted arc.
$$\angle AOB = \text{arc } AB.$$
So a 90° central angle cuts off a 90° arc (a quarter of the circle); a 180° central angle cuts off a semicircle. A central angle can be anything strictly between 0° and 360° (or between 0 and $2\pi$ radians) for a single arc, a full turn around the centre is $360°$ or $2\pi$ radians.
What Is the Central Angle Formula?
The central angle is found from the arc it cuts and the radius, and the formula depends on which unit you want.
In radians, the central angle is the arc length divided by the radius:
$$\theta_{\text{rad}} = \frac{s}{r},$$
where s is the arc length and r is the radius. This is the cleanest definition of radian measure: an angle of 1 radian cuts an arc equal in length to the radius.
In degrees, scale that ratio so a full circle reads 360°:
$$\theta_{\text{deg}} = \frac{s}{2\pi r} \times 360°,$$
since $2\pi r$ is the full circumference, and the fraction $\dfrac{s}{2\pi r}$ is the slice of the whole circle the arc represents. To move between the two units, use $180° = \pi$ radians, so $\theta_{\text{rad}} = \theta_{\text{deg}} \times \dfrac{\pi}{180°}$.
Quantity | Symbol | Meaning |
|---|---|---|
Central angle | $\theta$ | The angle at the centre, in degrees or radians |
Arc length | $s$ | Length of the intercepted arc along the rim |
Radius | $r$ | Distance from the centre to the rim |
Why the radian formula is just arc over radius
This is worth seeing once, because it explains why radians exist at all. Walk along the rim of the circle. As the angle at the centre opens, the arc you trace grows in exact proportion to it, and the constant of proportionality is the radius. Double the radius and the same angle traces twice the arc. So arc length is $s = r\theta$ when $\theta$ is in radians, and rearranging gives $\theta = s/r$. The radian is defined so that this relationship has no extra constant, which is exactly why it is the natural unit for circles.
The Central Angle Theorem
A central angle is also the anchor for one of the most-used circle results. The central angle theorem says that the central angle is twice any inscribed angle that opens onto the same arc.
An inscribed angle is an angle whose vertex sits on the circle (not at the centre) and whose sides are chords. If a central angle and an inscribed angle both intercept the same arc AB, then:
$$\angle AOB = 2 \times \angle ACB,$$
where O is the centre and C is a point on the circle. Equivalently, the inscribed angle is half the central angle. This is why a central angle is genuinely useful, it gives a direct line to angles drawn from the rim, which is the workhorse of circle-theorem problems and the reason an angle inscribed in a semicircle is always 90°.
A short reason the factor of 2 appears
Draw the radius OC. The triangle OAC is isosceles (OA and OC are both radii), so its two base angles are equal. The central angle for that triangle is an exterior angle, and an exterior angle equals the sum of the two opposite interior angles, two equal copies of the inscribed angle. That doubling is the whole source of the factor of 2.
Examples of the Central Angle in Geometry
With the definition, both formulas, and the theorem in hand, here are central angles being found. The problems move from a direct slice up to converting units and applying the theorem.
Example 1 - A pizza is cut into 8 equal slices. What central angle does each slice make at the centre?
The full circle is 360°, shared equally among 8 slices:
$$\theta = \frac{360°}{8} = 45°.$$
Final answer: each slice has a central angle of 45° (which is $\tfrac{\pi}{4}$ radians).
Example 2 - An arc of length 8 cm sits on a circle of radius 4 cm. Find the central angle in radians, then in degrees
Wrong attempt. A student writes $\theta = \dfrac{r}{s} = \dfrac{4}{8} = 0.5$ radians, dividing radius by arc out of habit. Check that against the definition: a radian is the angle whose arc equals the radius, so an arc that is longer than the radius (8 cm arc, 4 cm radius) must give an angle bigger than 1 radian, not smaller. An answer of 0.5 radians is going the wrong way.
Correct. The formula is arc over radius, not the reverse:
$$\theta_{\text{rad}} = \frac{s}{r} = \frac{8}{4} = 2 \text{ radians}.$$
Convert: $2 \times \dfrac{180°}{\pi} \approx 114.6°$.
Final answer: 2 radians, about 114.6°.
Example 3 - A central angle of 120° sits on a circle of radius 6 cm. Find the length of the intercepted arc
Convert the angle to radians, then use $s = r\theta$:
$$\theta_{\text{rad}} = 120° \times \frac{\pi}{180°} = \frac{2\pi}{3}, \qquad s = r\theta = 6 \times \frac{2\pi}{3} = 4\pi \approx 12.57 \text{ cm}.$$
Final answer: the arc is $4\pi \approx 12.57$ cm long.
Example 4. Convert a central angle of 60° to radians
Multiply by $\dfrac{\pi}{180°}$:
$$60° \times \frac{\pi}{180°} = \frac{\pi}{3} \approx 1.047 \text{ radians}.$$
Final answer: $\dfrac{\pi}{3}$ radians.
Example 5 - An inscribed angle of 35° opens onto arc AB. What is the central angle on the same arc?
By the central angle theorem, the central angle is twice the inscribed angle:
$$\angle AOB = 2 \times 35° = 70°.$$
Final answer: the central angle is 70°.
Example 6 - A circular running track has radius 50 m. A runner covers an arc of 75 m. What central angle (in degrees) have they swept around the centre?
Find the angle in radians first, then convert:
$$\theta_{\text{rad}} = \frac{s}{r} = \frac{75}{50} = 1.5, \qquad \theta_{\text{deg}} = 1.5 \times \frac{180°}{\pi} \approx 85.9°.$$
Final answer: about 85.9°.
Why the Central Angle Matters
A central angle looks like a slice of a circle, but it is the quantity that turns a circle into something you can measure, divide, and navigate by.
Pie charts and data. Every wedge in a pie chart is a central angle sized to its share of the whole: a category worth 25% of the data gets a $0.25 \times 360° = 90°$ central angle. Reading a pie chart is reading central angles.
Clocks and gears. A clock face is divided by central angles, each hour mark is 30° apart ($360° / 12$). Gear teeth, turbine blades, and bicycle spokes are all spaced by equal central angles so the load shares evenly.
Sectors, arc length, and area. The central angle is the key that unlocks both the arc length and the area of a sector, the slice's area is $\tfrac{1}{2} r^2 \theta$ in radians. Engineers sizing a fan of solar panels or a curved bridge segment start from the central angle.
Navigation and astronomy. Latitude and longitude are central angles measured from the centre of the Earth; the angular separation between two stars is a central angle on the celestial sphere.
For a Grade 9 student, the central angle is where a circle stops being one round shape and becomes a thing you can carve into exact, measurable pieces, and where the link from an angle to a length first appears.
Where Things Go Sideways With the Central Angle
Mistake 1: Flipping the radian ratio to radius over arc
Where it slips in: A student remembers "angle, arc, radius go together" but not the order.
Don't do this: Write $\theta = r / s$.
The correct way: The radian measure is arc over radius, $\theta = s/r$. Sanity-check: an arc longer than the radius gives more than 1 radian; an arc shorter than the radius gives less than 1.
Mistake 2: Confusing the central angle with the inscribed angle
Where it slips in: Both angles open onto the same arc, and the student treats them as equal.
Don't do this: Set the central angle equal to the inscribed angle on the same arc.
The correct way: The central angle is twice the inscribed angle on the same arc, $\angle AOB = 2 \angle ACB$. Check where each vertex sits, centre means central, on the rim means inscribed.
Mistake 3: Mixing degrees and radians in one calculation
Where it slips in: The arc-length formula $s = r\theta$ is used with $\theta$ in degrees.
Don't do this: Plug 120° straight into $s = r\theta$ without converting.
The correct way: $s = r\theta$ requires $\theta$ in radians. Convert first ($120° = \tfrac{2\pi}{3}$), then multiply.
Key Takeaways
A central angle in geometry has its vertex at the centre of a circle and equals the measure of its intercepted arc.
The formula is $\theta = s/r$ in radians and $\theta = \dfrac{s}{2\pi r} \times 360°$ in degrees, with $180° = \pi$ radians linking the two.
The central angle theorem says the central angle is twice any inscribed angle on the same arc.
The most common mistake is flipping the ratio to radius over arc, sanity-check against "arc longer than radius means more than 1 radian."
Central angles power pie charts, clocks, sectors, and navigation, anywhere a circle is divided into measurable pieces.
Practice These Problems to Solidify Your Understanding
A circle is divided into 5 equal sectors. Find the central angle of each.
An arc of length 12 cm lies on a circle of radius 3 cm. Find the central angle in radians.
An inscribed angle of 48° opens onto an arc. Find the central angle on the same arc.
Answer to Question 1: $360° / 5 = 72°$ each. Answer to Question 2: $\theta = 12 / 3 = 4$ radians. Answer to Question 3: $2 \times 48° = 96°$. If Question 2 gave you $0.25$ radians, recheck the order of the ratio (see Mistake 1).
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