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Ellipse: area and perimeter
Computes the area and perimeter of an ellipse
2024.Mar.28 18:45:19
a, b m ([L])   (ba) Semiaxes: major (a) and minor (b). •
θf ° Final angle for arc (from 0). •
Δθ rad Increment (for numerical integration). •
Show values Shows the coordinates of the graph. •

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: ba, 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) (xa)² + (yb)² = 1; (parametric) x = a cosθ, y = b sinθ .

The eccentricity, e, corresponds to e² = 1 − (ba)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)

References: Plate: EllipsePerimeter

• Wikipedia: Ellipse

Chadrupatla, T. R., and T. J. Osler, 2010, The perimeter of an ellipse.pdf, Math. Scientist, 35:122–131 (=),

• Composite Simpson's rule.

Area of an elliptical sector (Casio Computer Co., Ltd.).

Perimeter of an Ellipse (Numericana).

Unicode characters (Symbol, Math)

• 1894-01-01: Bose, Satyendranath (1974-02-04).

 
 
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Created: 2016-01-01 — Last modified: 2021-04-05