Chord Length CalculatorFree online math calculator — instant results, no sign-up
Calculate a circle chord from its central angle.
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About the Chord Length Calculator
The Chord Length Calculator finds the straight-line distance between two points on a circle using its radius and the central angle between them. Enter the angle in degrees to calculate the chord rather than the curved arc connecting the points. With the default radius of 5 and central angle of 90 degrees, the result is 7.0710678119.
- Radius
- The distance from the circle’s center to its circumference, entered in any consistent length unit, with a default of 5.
- Central angle (degrees)
- The angle at the circle’s center between the radii drawn to the chord’s endpoints, measured in degrees, with a default of 90.
Formula and method
c = 2r sin(θπ/360); with the default inputs, c = 2 × 5 × sin(90π/360) = 7.0710678119, rounded to 10 decimal places.
- cChord length, in the same length unit as the radius.
- rRadius of the circle.
- θCentral angle in degrees.
- πThe circle constant pi; π/180 converts degrees to radians, so θπ/360 is the half-angle in radians.
Bisecting the central angle creates two congruent right triangles, each with hypotenuse r and opposite side c/2. The sine relationship gives c/2 = r sin(θ/2). The calculator converts this half-angle to radians, evaluates its sine, and multiplies by twice the radius.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Radius5
- Central angle (degrees)90
Results
- Result7.0710678119
How to use the Chord Length Calculator
- 1Enter the circle’s radius in the Radius field; leave it at 5 to reproduce the default example.
- 2Enter the angle between the two endpoint radii in Central angle (degrees); use 90 for the default example.
- 3Use a positive radius and a central angle between 0 and 360 degrees for the intended circle geometry.
- 4Calculate the chord length using c = 2r sin(θπ/360).
- 5Read the result: the default inputs give 7.0710678119 in the same length unit as the radius.
Reading your result
The Result row reports the straight-line chord length, not the arc length or the circle’s circumference. For radius 5 and central angle 90 degrees, Result = 7.0710678119. No length-unit conversion is involved: the output uses whichever unit you chose for the radius.
Limitations
The calculation assumes a perfect circle in flat, Euclidean geometry. Use a positive radius and an angle from 0 to 360 degrees; at either endpoint of that angle range, the chord has zero length. The longest possible chord is the diameter, reached at 180 degrees. Angles outside this range may produce periodic or negative formula values that do not represent the intended chord length. Displayed decimals are rounded and do not imply that input measurements are equally precise.
Common questions
Why does a radius of 5 and a central angle of 90 degrees give 7.0710678119?+
The half-angle is 45 degrees, so the chord is 2 × 5 × sin(45°) = 5√2, which rounds to 7.0710678119.
Is the chord length the same as the arc length?+
No. A chord is the straight segment between the endpoints, while an arc follows the circle’s boundary. This calculator uses the sine formula for the chord, not the arc-length formula rθπ/180.
Can I enter a central angle greater than 180 degrees?+
Yes, geometrically a reflex central angle up to 360 degrees defines the same chord as its complementary full-turn angle, 360° − θ. The chord decreases from the diameter at 180 degrees to zero at 360 degrees.
Should I enter the central angle or an inscribed angle?+
Enter the central angle, whose vertex is at the circle’s center. If you know an inscribed angle intercepting the same arc, double that angle before entering it because the corresponding central angle is twice the inscribed angle.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.