Chord Length Calculator

What Is a Chord of a Circle?

A chord is a straight line that joins two points on a circle. The longest chord goes through the centre: it is the diameter, 2R. Every other chord is shorter, and cuts the circle into a small and a large circular segment.

Three measurements fix a chord once you know the radius: the central angle α between the two radii to its ends, the chord height h from its midpoint up to the arc, and the apothem a from the centre down to its midpoint. Height and apothem always add up to the radius: h + a = R.

Chord Length Formulas

All three describe the same right triangle, so they always agree: R = 10 and h = 4 means a = 6, and both give a chord of 16.

Why c = 2R·sin(α ÷ 2)

The apothem cuts the chord, and the central angle, exactly in half. That leaves a right triangle with the radius R as its longest side, the apothem a along the bottom and half the chord, c ÷ 2, opposite the angle α ÷ 2.

1. Sine of the half angle: sin(α ÷ 2) = (c ÷ 2) ÷ R, so c = 2R·sin(α ÷ 2).
2. Pythagoras: (c ÷ 2)² + a² = R², so c = 2√(R² − a²).
3. Height: a = R − h, so R² − a² = 2Rh − h² and c = 2√(2Rh − h²).

Chord Length From the Central Angle

Halve the angle, take its sine and multiply by the diameter (2R):

A 60° chord is exactly one radius long, which is why six of them fit around a circle as a regular hexagon. Angles over 180° give the same chord as 360° minus the angle: 240° and 120° both give R√3.

Chord Length From the Height

The chord height, also called the sagitta, is the distance from the middle of the chord to the arc. It is easy to measure on a real curve: lay a straight edge across it and measure the gap in the middle.

A height of more than R means the chord cuts off the larger part of the circle. The height can be at most 2R, where the chord shrinks to a point at the bottom.

Chord Length From the Apothem

The apothem is the shortest distance from the centre to the chord. Square the radius, take away the apothem squared, then take the square root and double it:

An apothem of 0 means the chord runs through the centre, so it is the diameter. The apothem must be less than R: at a = R the chord only touches the circle.

Radius From a Chord and Its Height

Turn the height formula around to find the radius of an arch, a curved wall or a broken plate from its span c and rise h:

Common Mistakes

• Forgetting to halve the angle. 2R·sin(100°) is not the chord; use sin(50°).
• Calculator in radian mode. sin(50) in radians is about −0.26, not 0.77.
• Mixing up chord and arc. The arc is the curved path and is always longer than the chord.
• Using the diameter as R. R is half the width across the circle.

Key Terms

Chord (c)

A straight line between two points on a circle.

Central angle

The angle α at the centre between the radii to the chord ends.

Height (h)

Chord midpoint to the arc, also called the sagitta.

Apothem (a)

Centre to the chord midpoint. a = R − h.

Segment

The area between a chord and its arc.

How to Use the Chord Length Calculator

1. Choose what you know. Pick Central angle, Height or Apothem, whichever you can measure besides the radius.
2. Enter the radius and the value. Type the radius, such as 10, and pick a unit if you want one. Then type the angle in degrees, the chord height or the apothem. Decimals and fractions such as 3/4 work.
3. Read and copy the result. The chord length appears as you type, with a drawing, the working, the other two measurements, the arc length and the segment area. Press Copy to copy it, or Clear or Esc to start again.

Example: a circular table has a 60 cm radius and you want to cut off a 15 cm deep flat edge. Choose Height, enter 60 and 15 and pick cm. The cut is 30√7 ≈ 79.372539 cm long.

Who Can Use This Calculator?

Frequently Asked Questions (FAQs)