Central Angle Calculator
Find a circle's central angle from an arc length or a chord length, using θ=s/r or θ=2·arcsin(c/2r).
📐 What is the Central Angle Calculator?
This central angle calculator finds a circle's central angle θ from either an arc length (θ=s/r) or a chord length (θ=2·arcsin(c/2r)). Choose a calculation mode, enter the radius and your known measurement, and it returns the angle in both degrees and radians.
The arc-length relation is the defining formula of the radian itself: a central angle of exactly 1 radian subtends an arc equal in length to the radius.
This calculator validates your input against the circle's physical limits, an arc length cannot exceed the full circumference and a chord length cannot exceed the diameter.
This calculator is useful for geometry students, and for anyone working with circular motion, sector area, or gear and pulley wrap-angle problems.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Right angle from arc length
Example 2 - Right angle from chord length
Example 3 - Small angle from arc length
❓ Frequently Asked Questions
🔗 Related Calculators
What is a central angle?
A central angle is an angle formed at the center of a circle by two radii drawn to two points on the circle's circumference. It directly determines the length of the arc and the chord connecting those two points.
What is the formula for the central angle from arc length?
θ = s/r, where s is the arc length and r is the radius, giving θ in radians. This is the defining relationship of the radian: an angle of 1 radian subtends an arc exactly equal in length to the radius.
What is the formula for the central angle from chord length?
θ = 2·arcsin(c/2r), where c is the chord length and r is the radius. This comes from splitting the isosceles triangle formed by the two radii and the chord into two right triangles.
Why does the central angle formula for arc length give radians, not degrees?
Because the radian is defined precisely as the ratio of arc length to radius (θ=s/r), the result is automatically in radians. Multiply by 180/π to convert to degrees, which this calculator does automatically.
What is the maximum possible arc length or chord length for a given radius?
The maximum arc length is the full circumference, 2πr (corresponding to a 360° central angle), and the maximum chord length is the diameter, 2r (corresponding to a 180° central angle, a straight line through the center). This calculator validates inputs against these limits.
How is a central angle different from an inscribed angle?
A central angle has its vertex at the circle's center, while an inscribed angle has its vertex on the circle itself, both angles can subtend the same arc, but the Inscribed Angle Theorem states the inscribed angle is always exactly half the corresponding central angle.
Can the central angle be greater than 180°?
Yes, if you consider the 'major arc' (the longer way around the circle) rather than the shorter arc between two points, the central angle for that major arc is 360° minus the central angle of the minor arc. This calculator computes the angle corresponding to whichever arc length or chord you enter.
What is a real-world use for the central angle formula?
Central angle calculations are used in circular motion problems, sector and segment area formulas, gear and pulley design (belt wrap angle), and dividing a circle into equal sections (like a pie chart or a clock face).
Is there a direct relationship between a central angle and its sector area?
Yes, sector area = (θ/2)r² when θ is in radians. Once you know the central angle from this calculator, you can plug it directly into that formula to find the sector's area.
Does this calculator work in both degrees and radians?
The inputs (radius, arc length, chord length) are simple lengths with no angle unit involved. The output is reported in both degrees and radians automatically, so you always see both.