Computes the area and perimeter of
a part of an ellipse ("upper" half), from angle 0 to a final given
θf. It is assumed that: b ≤ a,
and θf ≤ 180° (other values being related
by symmetry.
As there is no analytical form for the perimeter,
a numerical integration is done by the Simpson's rule,
from the parametric equation of the ellipse: (Cartesian)
(x⁄a)² + (y⁄b)² = 1;
(parametric) x = a cosθ,
y = b sinθ .
The eccentricity, e, corresponds to e² =
1 − (b ⁄ a)² . The numerical integral
for the length, L (from the general case of the length of an arc) is:
L = ∫0θf
√ (1 − e² cos²θ)
dθ
A plot is shown for the ellipse, and for the area and perimeter
from 0 to 180°.
Other suggested data:
(1) a = 1, b = 1 (circle) and
θf = 90°, giving A = π ⁄ 4 ≅ 0.7854 and
Per = π ⁄ 2 ≅ 1.5708;
(2) a = 1, b = 1 (circle) and
θf = 180°, giving A = π ⁄ 2 ≅
1.5708 and Per = π ≅ 3.1416. ellipse.xlsm (with macros) | 
