CalcKit
Circular Segment Calculator
Table of Contents
Circular segment definition
A circular segment is the part of a circle cut off by a straight line, between a chord and the arc above it. The chord and the arc meet at two points on the circle. It is the shape of a slice cut from a round table top, an arched window or a partly filled round tank seen from the end.
All its measures follow from any two of them: for example the chord and the height, or the radius and the angle. The angle is in degrees.
Parts of a circular segment
The seven main parts of a circular segment are height (h), chord length (a), arc length (b), radius (r), angle (α), area (A) and perimeter (P).
Height (h)
The height (or sagitta) of a circular segment is the distance from the midpoint of the arc to the midpoint of its chord. It is usually denoted by "h".
If you know the circle radius and the angle, you can find it with the following formula:
h = r * (1 - cos(α/2))
Chord length (a)
The chord of a circular segment is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is called a secant line.
The chord can be calculated using one of the following formulas:
-
If you know the circle radius and the angle:
a = 2 * r * sin(α/2) -
If you know the perimeter and arc length:
a = P - b
Arc length (b)
A circular arc is the arc of a circle between a pair of distinct points. The arc of a circle is defined as the part or segment of the circumference of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.
The formula to calculate the arc length of a circular segment is given by:
-
If you know the radius and angle:
b = r * α(when α is in radians)b = r * α * (π / 180)(when α is in degrees) -
If you know the perimeter and chord length:
b = P - a
Radius (r)
The radius of a circular segment is a straight line from the center of the circle to the circumference. It is usually denoted by "r".
It can be found with the following formula:
r = h / 2 + a² / 8 / h
Angle (α)
The angle of a circular segment is the measure of the arc in degrees. It is usually denoted by "α".
The formula to calculate the angle of a circular segment is given by:
-
If you know the chord length and the radius:
α = 2 * asin(a / 2 / r)(in degrees) -
If you know the height and the radius:
α = 2 * acos(1 - h / r)(in degrees)
Area (A)
The area of a circular segment is the region enclosed by the circular segment.
The formula to calculate the area of a circular segment is given by:
A = 1 / 2 * r² * (α * π / 180 - sin(α)) (with α in degrees)
Perimeter (P)
The perimeter of a circular segment is the distance around the edge of the segment which is formed by a chord and an arc. It is usually denoted by "P".
The formula to calculate the perimeter of a circular segment is given by:
P = a + b
Example
A segment of a circle with radius 10 and an angle of 60°:
-
Height:
h = 10 * (1 - cos(30°)) = 1.34 -
Chord length:
a = 2 * 10 * sin(30°) = 10 -
Arc length:
b = 10 * 60 * π / 180 = 10.47 -
Area:
A = 1 / 2 * 10² * (1.0472 - 0.8660) = 9.059 -
Perimeter:
P = 10 + 10.47 = 20.47
Frequently asked questions
How do I find the chord length from the radius and height?
a = 2 * √(h * (2 * r - h)). For a radius of 10 and a height of 2, the chord is 2 * √(2 * 18) = 12.
How do I find the radius from the chord and height?
r = h / 2 + a² / (8 * h). A chord of 12 with a height of 2 comes from a circle of radius 1 + 144 / 16 = 10.
What is the difference between a segment and a sector?
A segment is cut off by a chord, a straight line. A sector is cut by two radii from the center, like a slice of pizza. Sectors have their own Circular Sector Calculator.