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Segment Area Calculator

Table of contents

What is a segment of a circle? ⌓What is a chord of a circle?Formulas for a segment of a circle areaThis segment area calculator can work as a chord length calculator as well!FAQs

Use this segment area calculator to quickly compute the area of a segment. It can also be used to find chord length and arc length. If you're unsure what a segment of a circle is or even what a chord of a circle is, don't feel embarrassed – just scroll down to find a few definitions and some self-explanatory images.

What is a segment of a circle? ⌓

Circle segment

If you want to understand what the segment of a circle is, try to imagine cutting part of a circle off with a single, straight cut. And that's it! You've just created two parts of the circle, and the smaller one is called the circular segment. A more formal mathematical definition says that:

A circular segment is a region bounded by a chord and the arc of a circle (of less than 180°)

If it's equal to 180°, then it's simply a half circle – semicircle. According to some definitions, the central angle doesn't need to be smaller than 180° – in that case, you can say that cutting a circle with a line gives you two segments: a major segment and a minor segment.

Circle segments, major and minor

Have a look at the picture below to help you visualize the difference between segment and sector, as those two names are sometimes confused:

Circle sector and segment

What is a chord of a circle?

Circle chord

A chord is a line that connects two points on a circle. The infinite line extension of a chord is called a secant. The special case of a chord is one that passes through the center of a circle – and that's the circle diameter, of course!

Formulas for a segment of a circle area

To find the circle segment area, you need to know at least two variables. In our segment area calculator, you'll find two popular formulas implemented:

measures in circular segment
  1. Formula given radius and central angle

    Asegment = 0.5 × r² × (α – sin(α))

    Where does this formula come from? You can look at the segment area as the difference between the area of a sector and the area of an isosceles triangle formed by the two radii:

    Asegment = Asector - Atriangle

    Knowing the sector area formula:

    Asector = 0.5 × r² × α (learn more about this in the sector area calculator)

    An equation for the area of an isosceles triangle, given arm and angle:

    Aisosceles triangle = 0.5 × r² × sin(α) (learn more about this in the isosceles triangle area calculator).

    You can find the final equation for the segment of a circle area:

    Asegment = Asector - Aisosceles triangle = (0.5 × r² × α) - (0.5 × r² × sin(α)) = 0.5 × r² × (α – sin(α))

  2. Formula given radius and height

    Asegment = r² × arccos((r-h)/r) - (r-h) × √(2 × r × h - h²)

    where h is the height of a segment, also known as sagitta.

    This formula may be useful when you need to calculate, e.g., the volume of water in a pipe that is not completely full.

This segment area calculator can work as a chord length calculator as well!

Let's find out how to use this segment area calculator. In our example, we want to find the area of the cross-section of a partially filled pipe:

  1. Input the circle radius. Assume our pipe radius is 5 in.
  2. Enter the second variable. Let's say that it's filled 3 inches high, so input that value into the height box.
  3. There you go, that's it! Now we know that our segment area is equal to 19.8 in². Additionally, we determined the chord length (9.17 in), arc length (11.6 in), and central angle (132.84°).

Why might one need to calculate the area of a segment?

Calculating the area of a segment is often needed in fields like engineering, architecture, and various forms of structural design.

How do I calculate the segment area?

You can calculate the segment area in three steps:

  1. Determine the radius of the circle.

  2. Calculate the central angle.

  3. Apply the segment area formula:

    0.5 × r² × (α – sin(α))

What is the segment area if the circle has a 5 cm radius?

Assuming the height is 2 cm, the segment area will be 11.18 cm². You can calculate it using this formula:

r² × arccos((r-h)/r) - (r-h) × √(2 × r × h - h²)

What is the difference between a minor segment and a major segment?

A minor segment is smaller than a semicircle, while a major segment is larger. The difference is based on whether the central angle is less or more than 180 degrees.

Circle segment with central angle, height, length, chord length and area.

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