Segment Area Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Oct 08, 2022

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°).
Hanna Pamuła, PhD
Radius (r)
in
Circle segment with central angle, height, length, chord length and area



Central angle (α)
deg
Height (h)
in
Arc length (L)
in
Chord length (c)
in
Segment area (A)
in²
Check out 23 similar 2d geometry calculators 📏
AreaArea of a rectangleArea of crescent… 20 more
People also viewed…

Arc length

Our arc length calculator can calculate the length of an arc of a circle and the area of a sector.

Car heat

The hot car calculator shows how fast a car's interior heats up during a summer day.

Coffee kick

A long night of studying? Or maybe you're on a deadline? The coffee kick calculator will tell you when and how much caffeine you need to stay alert after not sleeping enough 😀☕ Check out the graph below!

Substitution method

Substitution method calculator shows you step-by-step how to solve systems of linear equations using the substitution method.
Copyright by Omni Calculator sp. z o.o.
Privacy policy & cookies
main background