Chord Length Calculator – Find Chord in Circle
A chord is a line segment whose endpoints lie on a circle.
If a chord subtends a central angle \(\theta\) at the center of a circle of radius \(r\),
you can compute its length with a simple trigonometric relationship.
1) Chord from radius and central angle
Consider two radii drawn to the chord endpoints. The triangle formed is isosceles with sides \(r,r\),
and the angle at the center is \(\theta\). If you split the triangle into two right triangles, each has:
• hypotenuse \(r\)
• acute angle \(\theta/2\)
• half-chord length \(c/2\)
Therefore:
\[
\sin\left(\frac{\theta}{2}\right)=\frac{(c/2)}{r}
\quad\Longrightarrow\quad
c = 2r\sin\left(\frac{\theta}{2}\right).
\]
If your angle is in degrees, convert first:
\[
\theta_{\text{rad}}=\theta_{\text{deg}}\cdot\frac{\pi}{180}.
\]
2) Chord from distance to center
Let \(d\) be the perpendicular distance from the circle’s center to the chord.
The radius to the midpoint of the chord forms a right triangle with:
• hypotenuse \(r\)
• one leg \(d\)
• the other leg \(c/2\)
\[
\left(\frac{c}{2}\right)^2 + d^2 = r^2
\quad\Longrightarrow\quad
c = 2\sqrt{r^2 - d^2}.
\]
This is valid when \(|d|\le r\). If \(|d|=r\), the chord collapses to a point (tangent point).
3) Converting between \(d\) and \(\theta\)
The same split-triangle gives:
\[
d = r\cos\left(\frac{\theta}{2}\right)
\quad\Longleftrightarrow\quad
\theta = 2\arccos\left(\frac{d}{r}\right).
\]
4) Useful extra circle segment quantities
Many chord problems also ask for heights and areas:
-
Sagitta (segment height):
\[
s = r - d.
\]
This is the “height” of the circular segment from the chord up to the arc (along the perpendicular through the center).
-
Arc length (for central angle \(\theta\) in radians):
\[
\ell = r\theta.
\]
-
Sector area:
\[
A_{\text{sector}} = \frac{1}{2}r^2\theta.
\]
-
Circular segment area (sector minus isosceles triangle):
\[
A_{\text{segment}} = \frac{1}{2}r^2\left(\theta-\sin\theta\right).
\]
5) Quick sanity checks
- If \(\theta\to 0\), then \(c\to 0\).
- If \(\theta=\pi\) (180°), the chord is a diameter and \(c=2r\).
- If \(d=0\), the chord is a diameter and \(c=2r\).
- If \(d\to r\), the chord shrinks to \(c\to 0\).
Example
If \(r=10\) and \(\theta=60^\circ\), then:
\[
c = 2r\sin\left(\frac{\theta}{2}\right)
= 2\cdot 10\cdot \sin(30^\circ)
= 20\cdot \frac{1}{2}
= 10.
\]
(If you instead meant \(\theta=120^\circ\), then \(c=2\cdot 10\sin(60^\circ)=10\sqrt{3}\approx 17.32\).)
