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Sofia Vasilievna
StudyMathGeometry

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

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
R- radius
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 segmentCreative Commons Attribution/Share-Alike License 3.0 (Unported)
 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)Creative Commons Attribution/Share-Alike License 3.0 (Unported)
 Radius:
 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 heightCreative Commons Attribution/Share-Alike License 3.0 (Unported)
 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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