Introduction
A chord is a straight line that joins two points on a circle. The Chord Length Calculator measures how long that line is.
You can solve it two ways:
- Radius + central angle: uses the formula L = 2r × sin(θ/2)
- Radius + sagitta (arc height): uses the formula L = 2√(2rh − h²)
Type your numbers, pick your units (mm, cm, m, in, or ft), and the answer updates as you go. You also get the half-chord, sagitta, central angle in degrees and radians, arc length, apothem, and circumference.
The tool draws a labeled circle diagram and shows each math step, so you can see how the answer was found. This helps with geometry homework, woodworking, metal bending, arch design, and any job where you need to measure across a curve.
How to use our Chord Length Calculator
Pick a method, type in your circle's numbers, and the calculator gives you the chord length plus the half-chord, sagitta, central angle, arc length, apothem, and circumference, with a diagram and step-by-step math.
Calculation method: Click the tab that matches what you know. Use "Radius + Central Angle" if you know the angle at the center. Use "Radius + Sagitta (Arc Height)" if you know how tall the arc rises above the chord.
Radius (r): Type the distance from the center of the circle to its edge. It must be more than 0. Pick your unit: mm, cm, m, in, or ft.
Central Angle (θ): Type the angle at the center between the two radii that touch the chord's ends. Choose degrees or radians. It must be more than 0 and less than 360° (or 2π radians).
Sagitta / Arc Height (h): Type the straight distance from the middle of the chord up to the arc. It must be more than 0 and cannot be bigger than the radius.
Display results in: Choose the unit you want all the answers shown in. You can change it any time and the results update right away.
Calculate: Results update as you type, but you can click this button to refresh them. Click "Reset / Clear All" to go back to the starting values.
What Is a Chord of a Circle?
A chord is a straight line that joins two points on the edge of a circle. Think of it like a straight shortcut across the circle instead of going around the curve. The longest chord of any circle is the diameter, because it passes right through the center.
Chord Length Formulas
There are two easy ways to find chord length (L):
- From radius and central angle: L = 2r × sin(θ / 2)
- From radius and sagitta: L = 2√(2rh − h²)
Here r is the radius, θ is the central angle, and h is the sagitta.
Parts of the Circle You Need to Know
- Radius (r): the distance from the center of the circle to its edge.
- Central angle (θ): the angle at the center made by the two radii that touch the ends of the chord.
- Sagitta (h): also called arc height. It is the distance from the middle of the chord straight up to the curve.
- Arc length (s): the curved distance between the two chord ends, found with s = rθ (with θ in radians).
- Apothem: the shortest distance from the center to the chord. It always meets the chord at a right angle and cuts it in half.
Why the Formula Works
Draw a radius to each end of the chord. You get a triangle with two equal sides. Now drop a line from the center straight down to the chord. That line splits the triangle into two matching right triangles. In each one, the short side is half the chord, so half of L equals r × sin(θ/2). Double it and you get the full chord length.
Quick Facts
- A bigger central angle makes a longer chord, up to 180°.
- At 180°, the chord is the diameter, so L = 2r.
- After 180°, the chord gets shorter again.
- The sagitta can never be bigger than the radius.
- Two chords the same distance from the center are always the same length.
Where Chord Length Is Used
Builders and engineers use chord length to lay out curved roads, arches, bridges, and round tanks. Woodworkers use it to cut curved pieces. Machinists use it to space bolt holes evenly around a circle. Surveyors use it to measure across bends in land or water.