# Circular Segment Calculator

## Circular segment definition

A circular segment is a portion of a circle that is enclosed between a chord and an arc. The chord and the arc intersect at two points on the circumference of the circle.

## 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 chorc length:

`b = P - a`

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)`

• If you know the height and the radius:

`α = 2 * acos(1 - h / r)`

### 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))`

### 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`