Geometry
How To Calculate The Area Of An Ellipse
An ellipse is a circle that has been stretched along one axis, and its area is the circle area with one radius swapped for the other.
Quick Answer
A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis
- a
- Semi-major axis
- b
- Semi-minor axis
Multiply pi by the longest radius and the shortest radius. For an ellipse 20 cm long and 12 cm wide, a is 10 and b is 6, so the area is pi x 10 x 6 = 188.5 square centimetres.
What Is The Area Of An Ellipse?
The area of an ellipse is pi times the semi-major axis times the semi-minor axis. The semi-major axis is the distance from the centre to the furthest point on the edge, and the semi-minor axis is the distance to the closest point. Every other length in the problem is either a whole diameter, which you halve, or a distraction.
The formula is a circle area with one radius replaced by a different radius. Take the area of a circle of radius a and squeeze it along one axis until the radius in that direction becomes b, and the area falls in exact proportion. That is why pi times a times b is the whole story and why there is no square root, no eccentricity and no approximation term anywhere in it.
The most common source of error is diameter versus radius. A problem that says the ellipse is 20 cm long almost always means the full length, so a is 10 and not 20. Feeding the diameter into the formula doubles the answer, and the error compounds if you make it on both axes. Read the wording once more and commit to which one you have before you multiply.
The aspect ratio a over b tells you how stretched the shape is. A ratio of exactly 1 means a is equal to b and the ellipse has collapsed back into a circle. A ratio of 2 means the ellipse is twice as long as it is wide. Orbits in our solar system have ratios very close to 1, which is why planets travel on nearly circular paths, while ratios above about 5 describe shapes most people would call an oval rather than an ellipse.
The circumference is the hard part, and it is worth being clear about why. Unlike the area, there is no simple closed form for the perimeter of an ellipse. The usual practical formula is Ramanujan's second approximation, which is accurate to a few parts in a billion over almost the whole range of shapes. It is what this calculator uses, and it is the reason the perimeter of a 20 by 12 ellipse comes out around 51.05 rather than some clean multiple of pi.
The equivalent circle radius is a genuinely useful by-product: the radius of the circle that would enclose the same area, which is the square root of a times b. For a 20 by 12 ellipse that is just under 7.75, and it tells you the total enclosed space at a glance without comparing two different formulas.
Formula
A = pi x a x b
The circle area formula with the two radii allowed to differ.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Area | square units | Enclosed area of the ellipse. |
| a | Semi-major axis | units | Half the longest distance across. |
| b | Semi-minor axis | units | Half the shortest distance across. |
| pi | Pi | none | The circle constant, about 3.14159. |
e = sqrt(1 - b^2 / a^2)
Eccentricity is zero for a circle and approaches one for a very long, thin ellipse.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| e | Eccentricity | none | How far from circular the ellipse is, between 0 and 1. |
| a | Semi-major axis | units | Half the longest distance across. |
| b | Semi-minor axis | units | Half the shortest distance across. |
r = sqrt(a x b)
The radius of the circle with the same area as the ellipse.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Equivalent radius | units | Radius of the area-matched circle. |
| a | Semi-major axis | units | Half the longest distance across. |
| b | Semi-minor axis | units | Half the shortest distance across. |
How To Calculate The Area Of An Ellipse
- 1
Identify the two radii
Decide whether the problem gives diameters or radii. If it gives the full width and full length, halve both. The longer one is a and the shorter is b.
- 2
Multiply the two radii
Compute a times b. For a of 10 and b of 6 that is 60.
- 3
Multiply by pi
Take the product times 3.141592653589793. Sixty times pi is about 188.5.
- 4
Sanity-check against the circle
The answer must be smaller than pi times a squared and larger than pi times b squared. If it falls outside that range, the radii are swapped or a diameter slipped in.
Examples
Example 1: An ellipse 20 cm by 12 cm
- Full length
- 20 cm
- Full width
- 12 cm
| Step | Calculation | Result |
|---|---|---|
| Semi-major axis a | a = 20 / 2 | 10 cm |
| Semi-minor axis b | b = 12 / 2 | 6 cm |
| Substitute | A = pi x a x b | 3.141592653589793 x 60 |
| Ellipse area | A | 188.5 square cm |
Result: The ellipse encloses 188.5 square centimetres, against 314.2 for a full circle of 10 cm radius.
Example 2: Fitting an elliptical table into a room
- Table length
- 180 cm
- Table width
- 90 cm
| Step | Calculation | Result |
|---|---|---|
| Semi-major axis a | a = 180 / 2 | 90 cm |
| Semi-minor axis b | b = 90 / 2 | 45 cm |
| Ellipse area | A = 3.141592653589793 x 4050 | 12723.4502 |
Result: The table top covers 12723.4502 square centimetres, a little under 1.3 square metres, and seats comfortably fit around an ellipse of that size.
Example 3: An elliptical pond
- Semi-major axis
- 8 m
- Semi-minor axis
- 5 m
| Step | Calculation | Result |
|---|---|---|
| Substitute | A = pi x a x b | 3.141592653589793 x 40 |
| Ellipse area | A | 125.7 square metres |
| Equivalent circle radius | equivalent radius | 6.32 metres |
Result: The pond holds about 125.7 square metres of water, the same as a circular pond of about 6.32 metres radius.
Calculator
Ellipse area
188.4956
- Ellipse perimeter
- 51.054
- Aspect ratio
- 1.6667
- Equivalent circle radius
- 7.746
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Area Of An Ellipse calculator page.
Common Mistakes
Using diameters instead of semi-axes
The formula wants radii. A 20 by 12 ellipse has a of 10 and b of 6, not 20 and 12, and using the diameters quadruples the area.
Swapping a and b
Multiplication is commutative so swapping them does not change the area, but it does change the reported aspect ratio. Keep a as the larger so the eccentricity stays between zero and one.
Using 3.14 for pi on a large shape
On a 40 metre ellipse the difference between 3.14 and the full value of pi is more than 60 square metres. Carry pi to at least ten digits.
Assuming a and b are perpendicular distances from a focus
They are measured from the centre. The focus distances are related to eccentricity and are not the numbers this formula takes.
Confusing area with perimeter
The area has a clean closed form. The perimeter does not, and the approximation used here is Ramanujan's second. Do not expect the perimeter to come out as a neat multiple of pi.
Reporting square units as linear units
An area in square metres written as metres is a dimensional error, and it makes every downstream comparison meaningless.
FAQ
How do you calculate the area of an ellipse?
Halve the two outer dimensions to get the semi-major and semi-minor axes, then multiply pi by one and then the other. For a 20 by 12 ellipse that is pi x 10 x 6, which is about 188.5.
What is the difference between the major and minor axis?
The major axis is the longest line through the ellipse and the minor axis is the shortest, and both pass through the centre. The semi-major and semi-minor axes are half of those, measured from the centre to the edge.
Is an ellipse a stretched circle?
Yes, in the specific sense that scaling a circle's width while keeping its height unchanged produces an ellipse, and the area scales by exactly the same factor.
How do you find the perimeter of an ellipse?
There is no exact closed form. Ramanujan's second approximation, which this calculator uses, is pi times three times a plus b minus the square root of three a plus b times a plus three b, and it is accurate to a few parts in a billion.
What does an eccentricity of zero mean?
It means a equals b, so the ellipse is a circle. An eccentricity approaching one means the ellipse is very long and thin.
Why is the area of an ellipse smaller than a circle of the same length?
Because the width has been squeezed while the length stayed the same. The area falls in exact proportion to the width, which is why pi times a times b is less than pi times a squared whenever b is less than a.
References
- [1]Wikipedia, Ellipse — https://en.wikipedia.org/wiki/Ellipse
- [2]Wikipedia, Ramanujan approximation for ellipse circumference — https://en.wikipedia.org/wiki/Ellipse#Circumference
- [3]Khan Academy, Conic sections — https://www.khanacademy.org/math/geometry