Chord length is one of 6 circles formulas in the geometry section of this library, and it is used at high school level.
Why chord length works
The two radii out to the endpoints and the straight line joining them form an isosceles triangle. Splitting it down the middle produces a right triangle whose hypotenuse is the radius and whose angle at the centre is half the original, so half the straight line is the radius times the sine of that half angle.
What each symbol means
$r$ is radius and $\theta$ the smaller central angle.
Chord length: when it holds
Use a consistent angle mode; the sine form works directly in radians or degrees when the calculator mode matches.
When it stops applying
Both endpoints must lie on the same circle of that radius. For an ellipse, or for two points at unequal distances from the centre, the isosceles triangle behind the argument disappears and the result is simply wrong.
Chord length: a worked example
For $r=5$ and $\theta=60^\circ$, $c=10\sin30^\circ=5$.
The mistake to avoid
What people do: Taking the sine of the whole central angle instead of half of it.
Why it goes wrong: With a radius of 5 and a central angle of 60 degrees that returns about 8.66 rather than the correct 5.
Do this instead: Halve the angle before taking its sine, because the right triangle covers only half of the wedge.
Chord length: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is radius and $\theta$ the smaller central angle.
- Check the conditions before substituting. Use a consistent angle mode; the sine form works directly in radians or degrees when the calculator mode matches.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Circles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where chord length comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Equation of a Circle — the lesson behind this formula: read and build circle equations on the coordinate plane.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about chord length
What is the longest straight line across a circle?
The diameter. At a central angle of 180 degrees the sine reaches 1, and the formula returns twice the radius.
How do I recover the central angle from a known chord?
Divide the chord by twice the radius, take the inverse sine of that, and double the answer.
Is a chord shorter than the rim it cuts off?
Always, apart from the trivial case of a zero angle, because a straight line is the shortest route between the two endpoints.
How far is the chord from the centre?
That distance is the radius times the cosine of half the central angle, the other leg of the same right triangle: about 4.33 for a radius of 5 and a 60-degree angle.