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Ellipse Area Calculator

Ellipse area from its two semi-axes — A = π·a·b.

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How to use

  1. 1.Enter the semi-major axis (a) — half of the oval's longest full width — in the first box, in any unit.
  2. 2.Enter the semi-minor axis (b) — half of the shortest full width — in the second box, using the same unit.
  3. 3.Read the ellipse area instantly below, shown with the worked A = π·a·b formula, plus perimeter and eccentricity.

About Ellipse Area Calculator

The area of an ellipse is A = π·a·b, where a is the semi-major axis and b is the semi-minor axis — the two "half-widths" measured from the center. This ellipse area calculator returns that value the instant you type both numbers, and it also gives the perimeter (circumference) and the eccentricity so you can describe the whole oval from a single pair of inputs.

The most common mistake is entering the full width and height instead of the semi-axes. Each axis you type must be half of the corresponding full diameter: if your oval is 10 units wide overall, its semi-major axis a is 5, not 10. Enter semi-axes and the classic formula A = π·a·b works; enter full axes by accident and the area comes out four times too large. The formula is a natural generalization of the circle: a circle is just an ellipse with a = b = r, and π·a·b then collapses to the familiar πr².

The area is in square units of whatever unit you used for the semi-axes. Keep both axes in the same unit and the result is that unit squared — centimeters give square centimeters, meters give square meters, inches give square inches. The tool never converts units for you, so the math stays exactly π·a·b.

Unlike its area, an ellipse has no simple exact formula for its perimeter — the true value is an elliptic integral. This calculator reports the perimeter using Ramanujan's second approximation, C ≈ π(a+b)·(1 + 3h / (10 + √(4 − 3h))) with h = ((a − b)/(a + b))², which is accurate to many decimal places for ordinary ovals. When a = b the shape is a circle, h becomes 0, and the formula returns exactly 2πa — the true circumference of a circle — so the tool degrades gracefully to the circle case. The eccentricity, e = √(1 − (b/a)²) using the longer axis as a, tells you how stretched the oval is: 0 is a perfect circle and values approaching 1 are long and thin.

Ellipse area shows up widely. Landscapers and builders use it for oval flower beds, patios, tables, mirrors, and rugs; engineers use it for elliptical duct, tank, and pipe cross-sections; runners meet it in the shape of a running track infield; and it appears throughout astronomy, where planets orbit on ellipses. Students use it across geometry, trigonometry, and calculus. Everything here runs locally in your browser — nothing is uploaded — so results are instant and private.

A quick worked example: with a = 5 and b = 3, the area is π × 5 × 3 = 15π ≈ 47.12 square units, the eccentricity is 0.8, and the Ramanujan perimeter is ≈ 25.53 units.

Frequently asked questions

What is the formula for the area of an ellipse?
The area of an ellipse is A = π·a·b, where a is the semi-major axis and b is the semi-minor axis (the two half-widths from the center). Multiply the two semi-axes together, then by π. For example, a = 5 and b = 3 give 15π ≈ 47.12 square units. A circle is the special case a = b = r, where the formula becomes πr².
Do I enter the semi-axes or the full axes?
Enter the semi-axes — half of each full width. If the oval is 10 units wide and 6 units tall overall, the semi-major axis a is 5 and the semi-minor axis b is 3. Using the full axes by mistake makes the area four times too large, because A = π·a·b uses the half-widths, not the diameters.
How is the perimeter of an ellipse calculated?
An ellipse has no exact elementary formula for its perimeter, so this tool uses Ramanujan's second approximation, C ≈ π(a+b)(1 + 3h/(10 + √(4 − 3h))) with h = ((a − b)/(a + b))². It is accurate to many decimal places for typical ovals. When a = b the shape is a circle and the formula returns exactly 2πa.
What does the eccentricity of an ellipse mean?
Eccentricity e measures how stretched the oval is, from 0 to 1. It is e = √(1 − (b/a)²) using the longer semi-axis as a. A value of 0 means a perfect circle (a = b), while values near 1 mean a long, thin ellipse. For a = 5 and b = 3 the eccentricity is 0.8.
What units does the ellipse area come out in?
It is unit-agnostic. Enter both semi-axes in the same unit and the area is that unit squared: centimeters give square centimeters, meters give square meters, inches give square inches. The tool does not convert units, so keep a and b in one consistent unit.

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