Segment Area Calculator
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 selfexplanatory 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:

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))

Formula given radius and height
A_{segment}= r² * arccos((rh)/r)  (rh) * √(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 crosssection of partially filled pipe:
 Input the circle radius. Assume our pipe radius is 5 in.
 Enter the second variable. Let's say that it's filled 3 inches high, so input that value into the
height
box.  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°).