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# Circular segment

##### Can calculate area, arc length,chord length, height and perimeter of circular segment by radius and angle.
Anton2011-05-08 10:01:09

Circular segment - is an area of a circle which is "cut off" from the rest of the circle by a secant (chord).

On the picture:
L - arc length
h- height
c- chord
a- angle

If you know radius and angle you may use the following formulas to calculate remaining segment parameters:

#### Circular segment formulas

Area:
$A=\frac{1}{2}R^2(\alpha-\sin{\alpha})$ [1]
Arc length:
$L={\alpha}R$
Chord length:
$c=2{R}{\sin{\frac{\alpha}{2}}}$
Segment height:
$h={R}\left(1-{\cos{\frac{\alpha}{2}}}\right)$

Circular segment
Chord length:
Height:
Perimeter:
Arc length:
Area:

But if you don't know radius and angle you still can caluclate the segment parameters by chord length and segment height:

Segment (defined by chord and height)
Area:
Arc length:
Angle (degrees):
Perimeter:

Formula for segment radius by chord and height:
$R=\frac{h}{2}+\frac{c^2}{8h}$

Then, you can caluclate segment angle using the following formula:
$\alpha=2\arcsin{ \frac{c}{2R} }$

You may also use the following calculator to obtain segment area by its radius and height:
Area of circle segment by radius and height
Area:
Chord length:
Perimeter:
Arc length:
Angle (degrees):

This calculator evaluates angle by the following formula:
$\alpha=2\arccos\left(1-\frac{h}{R}\right)$
then it uses formule [1] to calculate the segment area.

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