Geometry: The Sagitta
in Circles
When the word Sagitta is mentioned, we would associate
sagitta with a small constellation between the constellations Aquila and
Cygnus. Sagitta is one of the smallest
constellations in astronomy. In
mythology, the Sagitta refers to an arrow, involved in a number of myths: (1) the arrow Hercules used to kill an eagle
to free Prometheus, (2) the arrow used by Apollo to avenge Asclepius’ death,
and (3) the arrow Eros used to shoot Zeus to make him fall in love with Ganymede.
But did you know that sagitta was referred to a length in
geometry. The sagitta is the length from
the center of a circular arc to its base (defined by the circle’s chord).
Variables:
r = radius of the circle
c = chord length
s = sagitta
θ = angle between radius and line that connects center to
end-chord line (see the diagram above)
Derivation:
Calculating the length of the sagitta
Given radius and angle:
cos θ = (r – s) / r
r * cos θ = r – s
s = r – r cos θ
s = r * (1 – cos θ)
Example: r = 5, θ = 60°
s = 5 * (1 – cos 60°)
s = 2.5
Given radius and chord length:
(r – s)^2 + (c/2)^2 = r^2
(r – s)^2 = r^2 – (c/2)^2
r – s = √(r^2 – (c/2)^2)
s = r - √(r^2 – (c/2)^2)
Example: c = 10, r =
6
s = 6 - √(6^2 – (10/2)^2)
s ≈ 2.68338
Sources:
“Sagitta (geometry)”.
Wikipedia. https://en.wikipedia.org/wiki/Sagitta_(geometry) Retrieved May 6, 2016
“Sagitta Constellation”
Constellation Guide.
Constellations: A Guide to the
Night Sky http://www.constellation-guide.com/constellation-list/sagitta-constellation/ Retrieved May 18, 2016
This blog is property of Edward Shore, 2016.
