Sagitta method

Derivation of the Sagitta method

The Sagitta method makes it possible to determine a circle radius using a section of a circle . This is always useful when the entire image of a circle is not available.

With the Sagitta method, two points ( and ) on a circular line are marked and the length of the resulting circular tendon is determined. Then a vertical line is drawn at the center of the chord and the height of the circle segment is determined. The unknown radius from point to forms a right triangle with the difference and . Using the Pythagorean theorem , the following can be calculated from the measurements: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle L}$ ${\displaystyle h}$ ${\displaystyle r,}$ ${\displaystyle A}$ ${\displaystyle M,}$ ${\displaystyle r-h}$ ${\displaystyle l/2}$ ${\displaystyle r}$

${\displaystyle \left({\frac {l}{2}}\right)^{2}+(r-h)^{2}=r^{2}.}$

The transformation

${\displaystyle r^{2}={\frac {l^{2}}{4}}+r^{2}-2rh+h^{2}}$

and expanding with and ${\displaystyle -r^{2}}$ ${\displaystyle +2rh}$

${\displaystyle 2rh={\frac {l^{2}}{4}}+h^{2},}$

after further forming delivers:

${\displaystyle r={\frac {l^{2}}{8h}}+{\frac {h}{2}}.}$

Historically, the height of the circle segment was referred to as the sagitta and the distance between the tendon and the center of the circle as the apothema . ${\displaystyle h}$ ${\displaystyle r-h}$

Applications

• Experimental physics (nuclei and particles)
• Measurements and simulations for a time projection chamber with GEM technology

Individual evidence

1. ↑ Floria Naumann: Use of bubble chamber images in schools on an increased requirement level. 2.2.3 Geometric reconstructions with the recordings. TU-Dresden, September 10, 2015, p. 21 , accessed on January 14, 2019 .