Ellipse Area and Perimeter Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The area of an ellipse is exactly π × a × b, where a and b are its semi-major and semi-minor axes. The perimeter has no simple exact formula: a very precise approximation (Ramanujan) is used. For a = 5 cm and b = 3 cm, the area is about 47.12 cm² and the perimeter about 25.53 cm.
Explanation
An ellipse is the generalization of a circle: instead of a single radius, it has two semi-axes, the semi-major axis a (the longer one) and the semi-minor axis b (the shorter one). Its area is calculated very simply, by an exact formula that directly generalizes that of the circle — indeed, when a = b, the ellipse becomes a circle of radius r and the formula π × a × b gives exactly π × r², the formula in our circle area and perimeter calculator. The perimeter, on the other hand, is much trickier: unlike the area, there is no simple closed formula (with only elementary operations) to compute it exactly — the exact value is expressed with a complete elliptic integral of the second kind, a mathematical object beyond the scope of a direct calculation. In practice, approximations are used: this calculator adopts the one proposed by the Indian mathematician Srinivasa Ramanujan in 1914, one of the most accurate known, with a relative error typically below 0.04% even for very elongated ellipses. As with the area of a regular polygon or a trapezoid, plane geometry here combines exact formulas and approximations: knowing which is which avoids assuming that a result shown with two decimals is necessarily an exact value.
Example: ellipse a = 5 cm, b = 3 cm
Inputs
Semi-major axis: 5 cm. Semi-minor axis: 3 cm.
Calculation
Area = π × 5 × 3 = 15π ≈ 47.12 cm² (exact). Perimeter ≈ π × (3×(5+3) − √((3×5+3)×(5+3×3))) = π × (24 − √(18×14)) = π × (24 − √252) ≈ π × 8.125 ≈ 25.53 cm (approximate).
Result
This ellipse has an exact area of about 47.12 cm² and an approximate perimeter of about 25.53 cm.
Frequently asked questions
Why is the area exact but not the perimeter?
The area of an ellipse follows directly from that of a circle by a simple scaling in one direction, which gives an exact, elementary formula (π × a × b). The perimeter depends on how the curve bends at each point, a quantity that does not simplify into a simple algebraic formula — it requires an elliptic integral, computable numerically but not expressible with elementary operations.
How accurate is Ramanujan’s approximation?
Very accurate for everyday use: its relative error stays below 0.04% even for strongly eccentric (very flattened) ellipses, and it becomes exact when a = b (the circle case). Other approximations exist (for example the arithmetic-geometric mean, slower to compute but even more accurate), but Ramanujan offers the best simplicity/accuracy trade-off for a direct calculation.
What happens if a and b are equal?
The ellipse then becomes a circle of radius a = b, and both formulas simplify: the area becomes π × r² and the approximate perimeter gives exactly 2π × r, the exact circumference of the circle — a good way to check the calculator's consistency.