About the Circle Chord Length Calculator
The Circle Chord Length Calculator finds the straight-line distance between two points on a circle's circumference, given the circle's radius and the central angle between those two points. It's useful whenever a problem specifies an angle and radius but asks for the straight segment connecting the two endpoints, rather than the curved arc between them.
How It Works
Enter the circle's radius and the central angle in degrees formed at the circle's center by the two points. The calculator converts the angle to radians, then applies the chord length formula, which comes from splitting the isosceles triangle formed by the two radii and the chord into two right triangles using a perpendicular bisector. The output is the straight chord length in the same units as the radius.
Formula & Methodology
To derive this by hand, draw the two radii to the chord's endpoints and the chord itself, forming an isosceles triangle with two sides of length r and an included angle theta. Dropping a perpendicular from the center to the chord bisects both the chord and the angle, creating a right triangle with hypotenuse r and angle theta/2. The chord's half-length is then r times sin(theta/2), so the full chord is twice that.
Examples
Default 60 Degree Angle
A circle with radius 6 and a central angle of 60 degrees gives a chord length of exactly 6, since sin(30 degrees) = 0.5 and 2 times 6 times 0.5 equals 6, so in this case the chord happens to equal the radius.
A 90 Degree Central Angle
A radius of 10 with a 90 degree central angle gives sin(45 degrees) of about 0.7071, producing a chord length of about 14.142, which is 10 times the square root of 2.
Advantages
- Automatically converts the angle to radians internally, so you can work entirely in degrees without a separate conversion step.
- Produces the correct chord length regardless of whether you enter the minor or major arc's central angle, since the formula is symmetric around 180 degrees.
- Gives an exact decimal answer instantly for a formula that otherwise requires drawing and solving a triangle by hand.
Common Mistakes
- Confusing chord length with arc length, which measures the curved distance between the same two points rather than the straight-line distance.
- Forgetting to halve the angle before taking the sine, since the formula's sin(theta/2) comes from the bisected triangle, not the full central angle.
- Assuming chord length scales the same way arc length does; chord length is bounded by the diameter no matter how large the angle gets, while arc length keeps growing.
Edge Cases to Watch For
- A central angle of 180 degrees gives the longest possible chord for that circle: it becomes a diameter, since sin(90 degrees) equals 1, giving c = 2r exactly.
- A central angle of 0 degrees gives a chord length of 0, since the two points coincide.
- Entering the major arc's central angle, over 180 degrees, instead of the minor arc's still produces the correct chord length, because sin(theta/2) and sin((360 degrees minus theta)/2) work out to the same value.
- Chord length is always less than or equal to the arc length between the same two points, since a straight line is always the shortest path between two points on a curve.
Common Use Cases
- Geometry students solving problems that give a circle's radius and central angle and ask for the straight-line distance between two points on the circumference.
- Engineers or designers calculating the span of a curved structural element, such as the straight distance across a circular arch or truss segment.
- Anyone verifying triangle-based circle geometry by hand, since the formula's derivation directly illustrates how to split an isosceles triangle into two right triangles.