Chord Length Calculator

Welcome to the chord length calculator, a quick and easy tool that allows you to calculate the length of a chord in geometry. Read on to discover more about the following topics:

• What is the chord length of a circle?
• What is the chord length formula?
• How to use this chord length calculator; and
• Arc length vs. chord length.

Apart from calculating the cord length, this tool can also determine other parameters of a circle, such as the area and the diameter, just in case you need them.

Now, are you ready to discover how to find the length of a chord in a circle? Let's get into it!

By now, we've all heard of areas, diameters, radii, and sectors, but what is a chord?

In a circle, a chord is a segment with both extremities placed at two points belonging to the same circumference. Unlike the diameter, a chord does not have to pass through the center of the circle: you can draw a chord between any two distinct points that are at an equal distance from the centroid of the figure. This distance is none other than the radius of said circle.

In the image below, you can see a lovely chord, c, as well as the:

• Radius, r;
• Arc length, L;
• Area, A;
• Central angle, α; and
• Chord height, h.

Okay, now we know what the length of a chord is in geometry, but the real question is, how do we calculate it?

The equation for chord length is quite simple. In fact, there are multiple! Here are the most common ones:

1. Formula based on chord height:

   chord length = 2 × √(2 × radius × chord height − chord height²)

2. Formula based on the distance between the center of the circle and the midpoint of the chord (apothem):

   chord length = 2 × √(radius² − apothem²)

3. Formula based on the central angle:

   chord length = 2 × radius × sin(α / 2)

As you can see, we will always need to know the radius plus one of these three variables to calculate the chord length of a circle.

But don't worry! If you don't know the radius, you can use any of these formulas:

• Given the area: r = √(A / π)

• Given the circumference:

  r = C / 2π

• Given the diameter: r = d / 2

• Given the arc length and the central angle: r = L / α
  The central angle is expressed in radians.

• Given the area of the sector and the central angle:

  r = √(2 × Aₛₜ / α)

  The central angle (α) is expressed in radians.

• Given the area of the segment and the central angle:

  r = √[2 × Aₛₘ / (α − sin(α)]

  The central angle (α) is expressed in radians.

Hey, but now it's getting complicated, isn't it? Why don't you use our chord length calculator instead? 😎

Although it is possible to calculate the chord length manually using the chord length formula, our chord length calculator is a simple and intuitive tool that is guaranteed to make this task easier!

Here is what you need to do to calculate the length of a chord in geometry using this calculator:

1. Insert or calculate the radius.

   • If you know the radius, it's easy: type it in.

   • If you don't know it, check the Show other parameters box. Additional fields will appear, allowing you to calculate the radius in no time. (The Radius field will be populated automatically.)

2. Insert the central angle, α, the chord height, h, or the apothem, r − h. (See the image in the calculator for reference.)

3. All done! As soon as you fill either of these three fields, the chord length calculator will immediately display your result.

4. Check the box underneath, if you haven't already, to see other essential parameters:

   • Area of the circle;
   • Circumference;
   • Diameter;
   • Arc length;
   • Sector area; and
   • Segment area.