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Properties of a Circular area
Properties of a Semi-Circular area
Properties of a Circular Sector
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Properties of a Circular segment

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

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 =
...or
h =
...or
d =
icon

Geometric properties:
φ (rad) =
h =
d =
Area =
Perimeter =
Arc length =
Chord length =
dc =
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Definitions

Geometry

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

shape geometry

For a circular segment with radius R and central angle φ, the chord length LABand 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}

shape geometry

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 LABthe 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.

See also
Properties of a Circular area
Properties of a Semi-Circular area
Properties of a Circular Sector
All Geometric Shapes tools