Ellipse Area Calculator
Compute the area of an ellipse using \(A = \pi a b\), from semi-axes or full diameters, with unit conversions, eccentricity, and equivalent circle radius.
Interactive ellipse area tool
Select how your ellipse is specified, choose length units, and the calculator will evaluate the area \(A = \pi a b\) and key derived quantities. The tool handles both semi-axes and full diameters and reports results in several area units.
Longest half-axis from center to edge.
Shortest half-axis from center to edge.
Both axes must use the same unit.
Formula for the area of an ellipse
An ellipse is defined by two characteristic lengths: the semi-major axis \(a\) (the longest half-axis) and the semi-minor axis \(b\) (the shortest half-axis). Its area is given by the simple but powerful formula
\(A = \pi a b.\)
If you instead know the full diameters (also called the full major and minor axes), usually denoted by \(W = 2a\) and \(H = 2b\), first convert to semi-axes:
\(a = \dfrac{W}{2}, \quad b = \dfrac{H}{2}, \quad A = \pi a b = \dfrac{\pi}{4} W H.\)
Ellipse vs. circle – a special case
A circle is just a special ellipse where the two semi-axes are equal: \(a = b = r\). In that case, the general formula reduces to the familiar circle area formula:
\(A = \pi a b = \pi r^2.\)
The calculator automatically detects when \(a\) and \(b\) are equal (within a small tolerance) and reports that you effectively have a circle.
Eccentricity and how “stretched” the ellipse is
Besides area, engineers and mathematicians often describe an ellipse using its eccentricity \(e\), a measure of how elongated it is. For an ellipse with \(a \ge b > 0\), the eccentricity is defined by
\(e = \sqrt{1 - \dfrac{b^2}{a^2}}.\)
Values of \(e\) range from
- \(e = 0\) for a perfect circle (\(a = b\)), to
- values very close to 1 for a highly stretched ellipse.
Bounding rectangle comparison
Any ellipse with semi-axes \(a\) and \(b\) fits exactly inside a bounding rectangle of width \(2a\) and height \(2b\). The area of this rectangle is
\(A_{\text{rect}} = (2a)(2b) = 4ab.\)
Comparing the ellipse area to its bounding rectangle:
\(\dfrac{A_{\text{ellipse}}}{A_{\text{rect}}} = \dfrac{\pi ab}{4ab} = \dfrac{\pi}{4} \approx 0.785.\)
So an ellipse covers about 78.5% of the area of its bounding rectangle, regardless of the specific values of \(a\) and \(b\).
Equivalent circle radius
Sometimes you may want to replace an ellipse by a circle with the same area. Setting the circle area \(\pi r^2\) equal to the ellipse area \(\pi a b\) gives:
\(\pi r^2 = \pi a b \quad \Rightarrow \quad r = \sqrt{ab}.\)
The calculator reports this equivalent radius, which is helpful when comparing elliptical ducts or tanks to circular ones with the same cross-sectional area.
Ellipse area – FAQ
Which values should I use for a and b?
Always use the semi-axes: distances from the center to the ellipse in each principal direction. If you start from full diameters (end-to-end measurements), divide them by 2 to obtain a and b before applying the formula. The calculator can do this conversion automatically when you choose the diameter mode.
Do the units of a and b matter?
They matter only in the sense that both must be the same. If a and b are in meters, area will be in square meters; if in centimeters, area will be in square centimeters, and so on. This tool lets you select the length unit once and then shows the area in a range of common square units (m², cm², mm², ft²).
Why is the area formula so simple compared with the perimeter?
The area formula \(A = \pi a b\) has a clean closed form because it comes from integrating a quadratic curve. By contrast, the perimeter (circumference) of an ellipse does not have a simple elementary expression and is usually approximated using series or special functions. This calculator focuses on area, where an exact formula is available.
Where is ellipse area used in practice?
Ellipse areas appear in structural and mechanical engineering (beam and shaft cross-sections), ventilation and piping (ducts and conduits), architecture (arched openings, domes), and physics (phase-space plots, orbital cross-sections). Accurately computing the area helps with sizing, flow capacity, and stress calculations.
Can I use this calculator for exam and design work?
Yes, the tool is designed to match standard textbook formulas and unit practices. It can confirm hand calculations and provide quick checks during design. For formal design submissions, you should still document your assumptions and formula derivations, but this calculator gives a reliable numeric reference.
Frequently Asked Questions
How accurate is this ellipse area calculator?
The area formula \(A = \pi a b\) is exact. The only rounding comes from displaying results to a limited number of decimal places and from floating-point arithmetic in your browser, which is more than sufficient for typical engineering, physics, and educational applications.
Can I mix units for a and b?
No. To avoid hidden conversion errors, the calculator assumes that both axes are expressed in the same length unit. Convert all measurements to a common unit (for example, meters) before entering them, or change the global unit selector and re-enter the values.
Does this calculator handle tilted ellipses?
Yes, as long as you know the lengths of the principal axes, any orientation in the plane gives the same area; rotation does not change area. The tool assumes you are supplying the semi-major and semi-minor axis lengths, not arbitrary chord lengths.