Ellipse Area and Perimeter

ScopeThe area and perimeter of an oval - an elliptical bed or border, a running-track lane, an oval tabletop or arch, an elliptical head footprint. Area = pi a b (exact, a=major/2, b=minor/2); the perimeter has no elementary closed form, so this uses Ramanujan pi[3(a+b) - sqrt((3a+b)(a+3b))], within a few parts per million up to about a 2:1 oval (54 ppm at b/a 0.3, 0.08% at 0.1, 0.34% at 0.01). A 10 x 6 oval covers 47.1 area with a 25.5 perimeter and 0.80 eccentricity; make the axes equal (8 x 8) and it collapses to a circle, 50.3 and 25.1 (exactly pi r^2 and 2 pi r). A partial (segment) area and an elliptic-integral perimeter are separate. A shop and layout aid; verify critical dimensions on the work.

Formula or tablea = major/2, b = minor/2; area = pi a b (exact); perimeter ~= pi[3(a+b) - sqrt((3a+b)(a+3b))] (Ramanujan); eccentricity = sqrt(1 - (b/a)^2).

Source and editionThe ellipse area pi a b and Ramanujan's perimeter approximation pi[3(a+b) - sqrt((3a+b)(a+3b))] (standard geometry; Ramanujan 1914), by name; public domain.

Free accessPure plane geometry, public; the major and minor axes are user-supplied measurements.

What governsEstimate. AHJ and licensed professional govern.

API field namesmajor_axis, minor_axis, area, perimeter, eccentricity

• Area exact area = pi a b, a and b the semi-axesplane geometry
• Ramanujan perimeter pi[3(a+b) - sqrt((3a+b)(a+3b))], a few ppm for ordinary ovals; not exactRamanujan 1914
• Circle limit equal axes give a circle (pi r^2, 2 pi r)geometry