This all-in-one online Central Angle Calculator performs calculations using a formula that relates the central angle of a circle with its radius and the arc length. You can enter the values of any two parameters in the fields of this calculator and find the missing parameter.
What is a Central Angle?
Formally, a central angle is defined as an angle formed by two radii of a circle that share an endpoint at the center of the circle. The portion of the circle between the two radii is called the intercepted arc, and the measure of the central angle is equal to the measure of this arc.
In mathematical notation, if point O is the center of a circle and points A and B are points on the circle, then angle ∠AOB is the central angle that subtends arc AB.
A central angle can be measured in degrees or radians. A full circle is 360 degrees or 2\(\pi\) radians. Therefore, a central angle that intercepts a quarter of a circle has a measure of 90 degrees or \(\pi\)/2 radians.
When working with central angles, especially in applied mathematics or physics, radians are often preferred due to their direct relationship with arc length and radius.
Central Angle Formula
The central angle can be expressed using the following straightforward formula:
$$\theta = \frac{s}{r}$$
where
• \(\theta\) is the central angle in radians,
• \(s\) is the length of the arc,
• \(r\) is the radius of the circle.
This formula arises from the definition of radian measure: one radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle’s radius. This very formula is used in our Central Angle Calculator.
Central Angle vs. Inscribed Angle
It’s important not to confuse a central angle with an inscribed angle, which also subtends an arc but has its vertex on the circle itself, not at the center.
The measure of an inscribed angle is exactly half the measure of the corresponding central angle that subtends the same arc. This relationship is useful in many geometric proofs and constructions.
Note that we don’t specify units of measure in our calculator. We assume that all the lengths are measured in the same length unit.
Related calculators
Check out our other geometry calculators such as Circle Calculator or Sector Area Calculator.