in Central angle (α)
deg
Height (h)
in
Arc length (L)
in
Chord length (c)
in
Segment area (A)
in²

# Segment Area Calculator

By Hanna Pamuła, PhD candidate

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? ⌓ 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. Have a look at the picture below to help you visualise the difference between segment and sector, as those two names are sometimes confused: ## What is a chord of a circle? A chord is a line which connects two points on the circle - you can think of it as the bowstring of an ideally curved bow. The infinite line extension of a chord is called a secant. The special case of a chord is one that passes through the centre 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: 1. Formula given radius and central angle

`A segment = 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:

` A segment = A sector - A triangle `

Knowing the sector area formula:

`A sector = 0.5 * r² * α `

And equation for the area of an isosceles triangle, given arm and angle (or simply using law of cosines)

`A isosceles triangle = 0.5 * r² * sin(α)`

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

` A segment = A sector - A isosceles 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. volume of a fluid in a pipe or in a circular tank, which is not completely full.

## 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 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°).
Hanna Pamuła, PhD candidate