# 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 rh}$${\ displaystyle l / 2}$${\ displaystyle r}$

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

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 . ${\ displaystyle h}$ ${\ displaystyle rh}$

## 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 .