Circle segment

Circle segment

A segment of a circle (including circular section ) is in the geometry of the partial surface of a circular area , of a circular arc and a chord is limited (as opposed to from a circular arc and two circle radii limited " circle sector / circular cutout ").

Names and characteristics

Sizes of the circle segment:

• α = central angle
• b = circular arc
• h = segment height
• r = radius
• s = tendon
• A = segment area
• M = center of the circle

The area of a circle segment can be calculated from the circle radius and the associated center angle. To do this, the area of ​​the corresponding circle sector and the isosceles triangle AMB shown in the sketch are determined . If the center angle is smaller than 180 °, you have to subtract this area (sector area minus triangle area). If the center angle is greater than 180 °, the areas must be added . If the center angle is exactly 180 °, the segment of the circle is a semicircle area and the area of ​​the triangle is 0. ${\displaystyle r}$ ${\displaystyle \alpha }$

In the formulas in the following table, angles are to be used in radians . The conversion of the measure of an angle from degrees to radians is done with the factor (see radians ). ${\displaystyle \pi /180^{\circ }}$

Formulas for the circle segment (all angles in radians )
Area	${\displaystyle A={\frac {r^{2}}{2}}\cdot \left(\alpha -\sin \alpha \right)}$ ${\displaystyle A={\frac {r\cdot b}{2}}-{\frac {s\cdot (r-h)}{2}}}$ ${\displaystyle A={\frac {{\frac {1}{2}}\arctan \left({\frac {2h}{s}}\right)\cdot (4h^{2}+s^{2})^{2}+hs\cdot (4h^{2}-s^{2})}{16h^{2}}}}$ ${\displaystyle A=r^{2}\cdot \arccos {\left(1-{\frac {h}{r}}\right)}-(r-h)\cdot {\sqrt {2rh-h^{2}}}}$ ${\displaystyle A=r^{2}\cdot \arcsin {\left({\frac {s}{2r}}\right)}-{\frac {s\cdot (r-h)}{2}}}$ ${\displaystyle A\approx {\frac {2}{3}}s\cdot h}$
radius	${\displaystyle r={\frac {4h^{2}+s^{2}}{8h}}}$ ${\displaystyle r={\frac {s}{2\cdot \sin {\frac {\alpha }{2}}}}}$ ${\displaystyle r={\frac {h}{1-\cos \left({\frac {\alpha }{2}}\right)}}}$
Circular tendon	${\displaystyle s=2r\cdot \sin \left({\frac {\alpha }{2}}\right)}$ ${\displaystyle s={\frac {2h}{\tan \left({\frac {\alpha }{4}}\right)}}=2h\cdot \cot \left({\frac {\alpha }{4}}\right)}$ ${\displaystyle s=2\cdot {\sqrt {r^{2}-(r-h)^{2}}}=2{\sqrt {2rh-h^{2}}}}$
Segment height	${\displaystyle h=r\cdot \left(1-\cos \left({\frac {\alpha }{2}}\right)\right)}$ ${\displaystyle h=r-{\sqrt {r^{2}-\left({\frac {s}{2}}\right)^{2}}}=r-{\frac {1}{2}}{\sqrt {4r^{2}-s^{2}}}}$ ${\displaystyle h={\frac {s}{2}}\cdot \tan \left({\frac {\alpha }{4}}\right)}$ ${\displaystyle h=r-\cos \left({\frac {360b}{4\pi r}}\right)r}$
Arc length	${\displaystyle b=r\cdot \alpha }$ ${\displaystyle b={\frac {\alpha \cdot (4h^{2}+s^{2})}{8h}}}$ ${\displaystyle b={\frac {\arctan \left({\frac {2h}{s}}\right)\cdot (4h^{2}+s^{2})}{2h}}}$ ${\displaystyle b=2\cdot r\cdot \arcsin \left({\frac {s}{2r}}\right)}$
Center angle	${\displaystyle \alpha \ =2\cdot \arctan \left({\frac {s}{2(r-h)}}\right)}$ ${\displaystyle \alpha \ =2\cdot \arccos \left(1-{\frac {h}{r}}\right)}$ ${\displaystyle \alpha \ =2\cdot \arcsin \left({\frac {s}{2r}}\right)}$ ${\displaystyle \alpha \ =2\cdot \arcsin \left({\frac {4hs}{4h^{2}+s^{2}}}\right)}$ ${\displaystyle \alpha \ =4\cdot \arctan({\frac {2h}{s}})}$
Centroid	${\displaystyle x_{s}={\frac {4}{3}}\cdot {\frac {r\cdot \sin ^{3}({\frac {\alpha }{2}})}{\alpha -\sin \alpha }},\qquad y_{s}=0}$ ${\displaystyle x_{s}={\frac {s^{3}}{12\cdot A}},\qquad y_{s}=0}$ Special case semicircle: ${\displaystyle x_{s}={\frac {4r}{3\pi }},\qquad y_{s}=0}$

Sagitta

The segment height is also called sagitta ( Latin for "arrow"), and the associated formulas can be derived using the Pythagorean theorem. The distance of the difference between radius and segment height forms with half of the circular chord a right-angled triangle with the radius as the hypotenuse . Thus, the following equation, which can then reshape accordingly . ${\displaystyle r^{2}=({\tfrac {s}{2}})^{2}+(r-h)^{2}}$

Similar geometric objects

The three-dimensional analog is a segment of a sphere .

Web links

• Eric W. Weisstein : Circle segment . In: MathWorld (English).

Individual evidence

1. ↑ Horst Stöcker: Handbook of mathematical formulas and computational science . Springer, 1998, ISBN 0-387-94746-9 .
2. ↑ Eric W. Weisstein : Sagitta . In: MathWorld (English).