Use a chord of a circle calculator when you need the straight-line distance between two points on a circle. The correct calculation depends on which measurements you know: radius and central angle, radius and perpendicular distance from the center, or radius and segment height.
A chord is not the curved arc between the points. It is the straight segment joining them. This distinction helps you choose the right formula.
What Is a Chord of a Circle?
A chord of a circle is a line segment with both endpoints on the circle. A diameter is a special chord that passes through the center, so it is always the longest chord in a given circle.
The perpendicular line from the center of a circle to a chord bisects the chord. In other words, it divides the chord into two equal halves. This creates two right triangles and provides the geometric basis for several chord formulas.
Chord vs. diameter
Every diameter is a chord, but not every chord is a diameter. If a circle has radius (r), its diameter is (2r). No chord in that circle can be longer than (2r).
As a chord moves closer to the center, its length increases. When its perpendicular distance from the center becomes zero, the chord passes through the center and becomes a diameter.
Chord vs. arc, secant, and sagitta
A chord is straight, while an arc is the curved portion of the circumference between the same endpoints. A secant is an entire line that intersects a circle at two points; the part inside the circle is a chord.
The sagitta, also called the segment height, is the perpendicular distance from the chord’s midpoint to the corresponding arc. It is not the same as the distance from the chord to the circle’s center. For the minor circular segment, these measurements satisfy (h=r-d), where (h) is the sagitta and (d) is the center distance. For a broader introduction to the geometry, see this guide to the properties of a circle chord.
How Do You Calculate the Chord of a Circle?
Calculate chord length with (c=2\sqrt{r^2-d^2}) when the radius (r) and perpendicular center distance (d) are known. If the radius and central angle (\theta) are known, use (c=2r\sin(\theta/2)). With radius and sagitta (h), use (c=2\sqrt{2rh-h^2}).
Using radius and distance from the center
When you know the circle radius and the shortest distance from the center to the chord, use:
[
c=2\sqrt{r^2-d^2}
]
Here, (c) is chord length, (r) is radius, and (d) is the perpendicular distance from the center to the chord. The value of (d) must fall between 0 and (r).
Using radius and central angle
When you know the angle formed by radii drawn to the chord’s endpoints, use:
[
c=2r\sin\left(\frac{\theta}{2}\right)
]
The central angle must correspond to the chord you are measuring. If (\theta) is entered in degrees, the calculator or trigonometric function must be in degree mode. For a (180^\circ) angle, the result is (2r), confirming that the chord is a diameter.
Using radius and segment height
When you know the radius and sagitta, use:
[
c=2\sqrt{2rh-h^2}
]
This form comes from substituting (d=r-h) into the center-distance formula. For the usual minor segment, the sagitta lies between 0 and (r). A sagitta close to zero produces a short chord, while (h=r) produces the diameter.
Which Chord Formula Should You Use?
Choose the formula that uses the measurements you already have. Do not substitute a sagitta for a center distance: although both are measured perpendicular to the chord, they start and end at different points.
| Known measurements | Quantity to find | Formula |
|---|---|---|
| Radius (r), center distance (d) | Chord (c) | (c=2\sqrt{r^2-d^2}) |
| Radius (r), central angle (\theta) | Chord (c) | (c=2r\sin(\theta/2)) |
| Radius (r), sagitta (h) | Chord (c) | (c=2\sqrt{2rh-h^2}) |
| Chord (c), sagitta (h) | Radius (r) | (r=\frac{c^2}{8h}+\frac{h}{2}) |
| Radius (r), chord (c) | Center distance (d) | (d=\sqrt{r^2-(c/2)^2}) |
| Radius (r), chord (c) | Central angle (\theta) | (\theta=2\arcsin(c/(2r))) |
All linear measurements must use the same unit. If radius is entered in centimeters, for example, the chord result is also in centimeters. Angles are dimensionless but must be handled consistently as degrees or radians.
Why Do the Chord Formulas Work?
The formulas work because a perpendicular line from the circle’s center bisects the chord. The radius, center distance, and half-chord form a right triangle, allowing chord length to be derived with either the Pythagorean theorem or trigonometry.
Derivation with the Pythagorean theorem
Let half the chord be (c/2). The radius (r) is the hypotenuse, and the perpendicular center distance (d) is the other leg:
[
\left(\frac{c}{2}\right)^2+d^2=r^2
]
Rearranging gives:
[
\frac{c}{2}=\sqrt{r^2-d^2}
]
Doubling both sides produces (c=2\sqrt{r^2-d^2}). The factor of 2 matters because the right triangle contains only half of the full chord.
Derivation with trigonometry
The perpendicular bisector also divides the central angle into two equal angles. In either right triangle:
[
\sin\left(\frac{\theta}{2}\right)=\frac{c/2}{r}
]
Multiplying by (2r) gives (c=2r\sin(\theta/2)). This explains why the formula uses half the central angle rather than the complete angle.
Chord of a Circle Worked Examples
Keep full precision during each calculation and round only the final answer.
Example 1: Radius and center distance
Suppose a circle has radius 10 cm and the chord is 6 cm from the center:
[
c=2\sqrt{10^2-6^2}
]
[
c=2\sqrt{100-36}=2\sqrt{64}=16\text{ cm}
]
The chord is 16 cm long. The result is valid because it is shorter than the 20 cm diameter.
Example 2: Radius and central angle
For a radius of 8 m and a central angle of (60^\circ):
[
c=2(8)\sin(30^\circ)=16(0.5)=8\text{ m}
]
The chord length equals the radius in this case. The two radii and the chord form an equilateral triangle.
Example 3: Radius and sagitta
For a radius of 13 cm and sagitta of 5 cm:
[
c=2\sqrt{2(13)(5)-5^2}
]
[
c=2\sqrt{130-25}=2\sqrt{105}\approx20.49\text{ cm}
]
The corresponding center distance is (13-5=8) cm. Substituting (r=13) and (d=8) into the center-distance formula produces the same result.
How Can You Find the Radius or Another Missing Measurement?
Chord relationships can be rearranged to find radius, center distance, sagitta, or central angle. A calculation is possible only when the known values define the circle unambiguously.
Finding radius from chord and sagitta
If chord length (c) and sagitta (h) are known, use:
[
r=\frac{c^2}{8h}+\frac{h}{2}
]
For a 12 cm chord with a 2 cm sagitta, (r=144/16+1=10) cm.
Finding center distance or central angle
Given radius and chord length, the perpendicular distance from the center is:
[
d=\sqrt{r^2-\left(\frac{c}{2}\right)^2}
]
The minor central angle is:
[
\theta=2\arcsin\left(\frac{c}{2r}\right)
]
The ratio (c/(2r)) must not exceed 1. If it does, the measurements describe an impossible chord because the proposed chord is longer than the circle’s diameter.
Common Chord Calculation Mistakes
Most incorrect results come from choosing the wrong input relationship rather than from the arithmetic itself. Check the meaning, unit, and valid range of every measurement before calculating.
- Confusing the sagitta with the perpendicular center distance
- Using the full central angle where the formula requires (\theta/2)
- Mixing centimeters, meters, inches, or other linear units
- Calculating only (c/2) and forgetting to double the result
- Using degree values while a trigonometric calculator is in radian mode
- Accepting impossible inputs such as (d>r) or (c>2r)
Round at the end of the calculation. Early rounding of a square root or sine value can noticeably alter the final chord, especially for large circles.
FAQ
What is the formula for the chord of a circle?
The formula is (c=2\sqrt{r^2-d^2}) when radius (r) and perpendicular center distance (d) are known. When radius and central angle are known, use (c=2r\sin(\theta/2)).
Can a chord be longer than the diameter?
No. The diameter passes through the center and is the longest possible chord, with length (2r). A calculated chord greater than (2r) indicates invalid inputs or an incorrect formula.
How do you find chord length without the central angle?
Use the radius and perpendicular center distance in (c=2\sqrt{r^2-d^2}). If you know the radius and sagitta instead, use (c=2\sqrt{2rh-h^2}).
How do you calculate radius from chord length and segment height?
Use (r=c^2/(8h)+h/2), where (c) is chord length and (h) is segment height or sagitta. Both measurements must use the same linear unit.
Are chord length and arc length the same?
No. Chord length measures the straight line between two points on a circle, while arc length follows the curved circumference between them. Except in the limiting case where both approach zero, the minor arc is longer than its chord.
Why does a perpendicular radius bisect a chord?
The radii drawn to the chord’s endpoints have equal length, and the perpendicular segment is shared by the two resulting right triangles. The triangles are congruent, so the two halves of the chord are equal
Liam Turner is a guitar theory writer and chord analysis specialist at Chord Calculator. He focuses on chord construction, music theory, scale relationships, harmony tools, and guitar learning resources for musicians, producers, songwriters, and beginners. Liam creates practical, easy-to-follow content around chord progressions, interval formulas, key analysis, and interactive music tools to help users better understand harmony, songwriting, and instrument practice.