## Properties of a Circular segment

This tool calculates the basic geometric properties of a circular segment. Enter below the circle radius R and either one of: central angle φ or height h or distance d. Note, that the angle φ can be greater than 180° which represents a segment bigger than the semicircle. In that case distance d is negative and height h is bigger than R. The calculated results will have the same units as your input. Please use consistent units for any input.

 R = φ = deg rad ...or h = ...or d = Geometric properties: φ (rad) = h = d = Area = Perimeter = Arc length = Chord length = dc =

## Definitions

### Geometry

For a circular segment the definitions shown in the following figure are used:

For a circular segment with radius R and central angle φ, the chord length LAB and its distance d from centre, can be found from the right triangle that occupies half of the region defined by the central angle (see next figure):

$\begin{split} & L_{AB} & = 2L_{MB} = 2 R \sin{\frac{\varphi}{2}}\\ & d & = R \cos{\frac{\varphi}{2}} \end{split}$

The height h of the circular segment h and its arc length L are found easily:

$\begin{split} & L & = \varphi R\\ & h & = R-d \end{split}$

The area A and the perimeter P of a circular segment, can be found with these formulas:

$\begin{split} A & = \frac{\phi-\sin{\varphi}}{2} R^2 \\ P & = L + L_{AB} \end{split}$

where L the arc length and LAB the chord length.

The centroid (center of gravity) of the circular segment is located along the bisector of the central angle φ, and at a distances from the chord equal to:

$d_c = \frac{4 R \sin^3{\frac{\varphi}{2}} }{3 \left(\varphi-\sin{\varphi}\right)}-d$

where d the distance of the chord from the centre.