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CalcMax

Ellipse Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

47.1239 cm²

Area

Perimeter
25.5270 cm
Eccentricity
0.8000

An ellipse calculator takes the two axes of an oval and returns three figures: the area it covers, the distance around its rim, and how far it departs from being a circle. The first two come back in centimetres — squared for the area, plain for the perimeter — and the third is a bare number with no unit at all, for reasons the page explains below. An ellipse is what a circle becomes when it is stretched further in one direction than the other, and the two axes are those two directions. The major axis is the longer one, running through the centre from end to end; the minor axis is the shorter one, at right angles to it. Half of each is called a semiaxis, and the distinction matters here because the standard formulas are written in semiaxes while this page asks for the full lengths: the two fields take what a tape measure would give you across a table top or a running track, and the halving happens inside. The area is the one figure with a formula everyone can use unaided. It is pi multiplied by the two semiaxes, which is the same expression as the area of a circle once you notice that a circle is just an ellipse whose two axes are equal. If your two measurements are equal, this page and the circle area page will agree exactly, and so will the rim length here and the circumference there — the first row of the table is there to be checked against both. The perimeter is the figure that needs the warning. Unlike the area, the distance around an ellipse has no formula that can be written down in a finite number of ordinary operations; it is an elliptic integral, and the only practical way to get a number is an approximation. The one used here is Ramanujan's second, which is accurate enough that the error never shows in the four decimals printed for any ellipse you could measure — but it is an approximation, and it behaves in a particular way at the two extremes. When the two axes are equal it is exactly right, because the two ends of the formula cancel. When the minor axis shrinks towards nothing the ellipse flattens towards a line segment, and the approximation comes out slightly low: a major axis of 10 and a minor axis of 0 should give a rim of 20 along the doubled semiaxis, and this page prints 19.992. The eccentricity is the third figure and the one nobody expects. It is a single number between 0 and 1 that says how far the ellipse is from being a circle: 0 is a circle, and the closer to 1 it gets, the flatter the ellipse. It has no unit because it is a ratio — it is the same number whether you measure in centimetres or in miles — and the page prints it with no unit for that reason. The three figures scale differently, and the table makes that visible in one pair of rows. Double both axes and the area goes up fourfold, the rim only doubles, and the eccentricity does not move at all, because a bigger ellipse of the same proportions is the same shape. That is why a photograph can be enlarged without changing what it depicts, and why the eccentricity alone is enough to describe the shape of an orbit while the axes are needed to describe its size. One thing the page will not do is straighten out your two measurements for you. If the shorter axis is entered in the longer box, it says so and stops rather than quietly swapping them: a page that reorders your numbers makes the figure on screen disagree with the label beside it, and you would have no way of knowing it had happened.

Common ellipses: the area, the perimeter and the eccentricity from the two axes

Major axis (cm)Minor axis (cm)Area (cm²)Perimeter (cm)Eccentricity
101078.539831.41590
10862.831928.36170.6
10647.123925.5270.8
2012188.495651.0540.8
10431.415923.01310.9165
10215.70821.010.9798
1017.85420.31960.995
100019.9921

Eight ellipses and five columns, with the two axes going in and three figures coming out. The first row is the circle and it is the row to check against other pages: an ellipse of 10 by 10 is a circle of radius 5, so its area of 78.5398 is what the circle area page prints and its perimeter of 31.4159 is what the circumference page prints. The second row is a gentler oval, 10 by 8, with an eccentricity of 0.6 — the shape of a good many real objects, from a racetrack to an oval table. The third row is the pair the page loads with, 10 by 6, whose eccentricity is exactly 0.8. The fourth row is that same ellipse with both axes doubled, and the three figures respond in three different ways — the area four times larger, the perimeter twice, and the eccentricity unchanged — which is the whole reason eccentricity is quoted separately from size. The fifth and sixth rows flatten the same 10 centimetre major axis further, to 4 and then to 2, and they show the eccentricity climbing towards its limit while the perimeter barely moves. The final row is the degenerate case where the ellipse has collapsed to a line segment: the area is zero and the eccentricity has reached its limit of 1, and the perimeter column shows 19.992 where the true value is exactly 20, which is the largest error the approximation makes anywhere in the table. Note that the last two rows have almost the same perimeter — 20.3196 and 19.992 — despite one being a thin ellipse and the other a straight line, because a rim length stops responding to the minor axis once the shape is thin enough.

Formula

A = πab P ≈ π(a + b)·(1 + 3h ÷ (10 + √(4 − 3h))) e = √(1 − b² ÷ a²)

a
Half the major axis — the semimajor axis. This page asks for the full major axis and halves it, because that is what a tape gives you; the formula is stated in semiaxes because that is how it is always written
b
Half the minor axis. When b equals a the ellipse is a circle, and the area formula collapses into the familiar πr² with r standing for either semiaxis
πab
The area. It is exact, unlike the perimeter below — multiply the two semiaxes and multiply by π and you have the answer to as many places as you care to work out
h
A measure of how unequal the two semiaxes are, computed as the square of their difference over their sum. It is 0 for a circle and approaches 1 as the ellipse flattens towards a line, and it is the quantity the perimeter approximation is built on
P
The perimeter of an ellipse, in centimetres. It has no elementary closed form, so this is Ramanujan's second approximation: exact when the two axes are equal and slightly low when the minor axis approaches zero
e = √(1 − b² ÷ a²)
The eccentricity, a pure number from 0 to 1 with no unit. It is 0 for a circle and rises towards 1 as the ellipse flattens, and it is the measure astronomers and opticians quote when only the shape matters and not the size
Four decimal places
How wide the readings are printed. The area is exact underneath; the perimeter is an approximation; and the eccentricity is a ratio, so on a nearly circular ellipse its leading digits carry most of the information

Anything oval that has to be measured is the plainest case: a round table that turns out not to be round, an oval mirror or rug or basin, a racetrack or a running track whose two straights and two bends are described by its axes, an oval conference table, a swimming pool with rounded ends, and a picture frame or mat with an oval cutout. In each of these the area is what a quantity of material or a coverage calculation needs, and the rim length is what a frame, a trim, a banding or a length of edging is bought by. Landscaping and construction use it for oval planting beds, curved paving, elliptical patios and any shape traced out with a string between two pins, which is the standard way an ellipse is marked out on the ground. Manufacturing uses it for elliptical handholes, oval tubing, flattened pipe, and the cross-section of anything drawn or pressed into an oval; the eccentricity matters there because two ovals with the same area can be noticeably different shapes and only one of them will fit a given fitting. In astronomy the eccentricity is the number that matters most: planetary orbits are ellipses with the Sun at one focus, and the eccentricity says how nearly circular each orbit is, with the Earth at 0.0167 and a comet far higher. Optics uses it for the elliptical flat mirrors that fold a light path in a telescope, where the axes of the mirror are set by the angle it is tilted at, and medicine uses it for the cross-sections of vessels and chambers and for the elliptical fields used in radiotherapy. In statistics it is the shape of a confidence ellipse, whose axes come from two standard deviations and whose area gives a coverage. In the classroom it is the demonstration that a circle is a special case of something more general — pin a loop of string around two drawing pins, trace it with a pencil, and the shape changes as the pins move apart while the perimeter stays exactly the same as the loop, which is a fact worth noticing in its own right. And in everyday arithmetic it answers whether an oval rug will cover a floor, how much cloth an oval tablecloth takes, and what length of edging an oval flower bed needs.

Worked examples

  1. An ellipse 10 across and 6 the other way

    1. Halve both: the semimajor axis is 5, the semiminor axis is 3
    2. Area: π × 5 × 3 = 47.1239 square centimetres
    3. Eccentricity: √(1 − 3² ÷ 5²) = √0.64 = 0.8
    4. Perimeter: h = ((5 − 3) ÷ (5 + 3))² = 0.0625, giving π × 8 × 1.0157… = 25.527 centimetres

    The pair the page loads with, chosen because the eccentricity comes out at exactly 0.8 — a number a reader can check by hand and a clean illustration of what the figure means. The steps start by halving, which is the step the two fields hide: multiplying 10 by 6 and then by π would give four times the area, and that is the commonest mistake with this page.

  2. The same ellipse doubled: 20 by 12

    1. Halve both: 10 and 6
    2. Area: π × 10 × 6 = 188.4956 — four times the previous example
    3. Eccentricity: √(1 − 36 ÷ 100) = 0.8, unchanged
    4. Perimeter: 51.054 — twice the previous example

    Every axis doubled, and the three figures respond in three different ways. The area goes up fourfold, the rim only doubles, and the eccentricity does not move at all. Nothing about the shape has changed, which is exactly what the eccentricity is saying — it describes proportion and not size, which is why it is the figure used to compare orbits.

  3. A circle in disguise: 10 by 10

    1. Halve both: the semiaxes are 5 and 5
    2. Area: π × 5 × 5 = 78.5398 square centimetres
    3. Eccentricity: √(1 − 25 ÷ 25) = 0
    4. Perimeter: the approximation becomes π × 10 × 1 = 31.4159 centimetres

    An ellipse with equal axes is a circle, and this row can be checked against two other pages. A 10 by 10 ellipse is a circle of radius 5, and 78.5398 is the area the circle area page prints for that radius, while 31.4159 is the circumference the circumference page and the two-way conversion page both print for it. The perimeter approximation is also exact here rather than close, which is one of the two reasons it was chosen.

  4. A very flat ellipse: 10 by 2

    1. Halve both: 5 and 1
    2. Area: π × 5 × 1 = 15.708 square centimetres
    3. Eccentricity: √(1 − 1 ÷ 25) = 0.9798
    4. Perimeter: 21.01 centimetres — about a twentieth more than the 20 that out-and-back along the major axis would be

    An eccentricity of 0.9798 is close to the limit, and this is the row that makes the point that the eccentricity is not a percentage of anything. A rim of 21.01 around a shape whose longest dimension is 10 shows how little distance a very flat ellipse adds over the straight line it is collapsing towards — five per cent, not half.

  5. Fully flattened: 10 by 0

    1. Halve both: 5 and 0
    2. Area: π × 5 × 0 = 0 — the shape has collapsed to a line segment and covers nothing
    3. Eccentricity: √(1 − 0 ÷ 25) = 1, the limit
    4. Perimeter: 19.992 centimetres — and this is where the approximation shows its one flaw

    The extreme the page is honest about. A minor axis of zero makes the ellipse a line segment of length 10, so the true rim is 20 — twice the semimajor axis, out and back. The approximation prints 19.992, a shortfall of 0.008. That is the whole cost of using a formula instead of an elliptic integral, and it is the largest error the approximation makes anywhere on the page.

Limitations

This page computes an ellipse from its two axes and nothing else: it does not take an area or a perimeter and recover the axes, it does not give the foci or the focal distance, it does not give the tangent or the normal at a point, and it does not draw anything. Both axes are required, and the shorter one must be entered in the minor axis box; put the longer figure there and the page reports that the longer of the two belongs in the major axis box rather than quietly swapping them round. The two are allowed to be equal, which is a circle, and both are allowed to be zero, which is a point. Neither may be negative. The perimeter is an approximation and not an exact figure — Ramanujan's second approximation, exact when the two axes are equal and slightly low as the minor axis approaches zero, where a major axis of 10 with a minor axis of 0 gives 19.992 against a true 20. For an ellipse you can actually measure, the error never reaches the fourth decimal, and there is no way to get a more accurate figure on this page because the exact perimeter is an elliptic integral. The eccentricity has no unit, because it is a ratio of two lengths and comes out the same whatever they were measured in; anything that prints a unit beside it is wrong. Both axes accept centimetres, metres, millimetres, inches or feet and both outputs in centimetres come back that way whatever was entered, because the conversion happens on the way in; an area measured in inches converts by the square of the length factor, so it is 6.4516 and not 2.54. Four decimal places is a display width rather than a claim of precision, and it is worth remembering that on a nearly circular ellipse the eccentricity is small and its leading digits carry nearly all of the information. Nothing here handles an ellipse that is rotated, and nothing handles an oval that is not a true ellipse — an egg shape and a stadium shape both look oval and neither is one.

Frequently asked questions

Are the two boxes the full axes or the half axes?
The full axes — whatever a tape measure gives you from one end of the oval to the other. The formulas use half of each, so the page halves them for you, and writing the halving into the boxes would mean asking you to divide your own measurements by two before typing them in. This is the only step in the arithmetic that is hidden, which is why every worked example starts by naming it.
Why is the perimeter not exact?
Because the distance around an ellipse has no formula that can be written out in ordinary operations — it is an elliptic integral, and there is no way around that. What the page uses is Ramanujan's second approximation, which is exact when the two axes are equal and a little low as the ellipse flattens; at the extreme, a major axis of 10 with a minor axis of 0 gives 19.992 where the true answer is 20. For any ellipse you could measure with a ruler, the difference never reaches the fourth decimal printed.
What does the eccentricity actually tell me?
How far the ellipse is from being a circle, as a single number between 0 and 1. Zero is a perfect circle and the number rises as the shape flattens. It has no unit because it is a ratio of two lengths, so it is the same figure whether you measured in centimetres or in miles — which is why it is the number used to compare the shapes of planetary orbits, whose sizes are wildly different and whose shapes are not.
Why did the area change when I doubled the axes but the eccentricity did not?
Because the eccentricity describes the shape and the area describes the size, and doubling both axes changes only one of them. Double both and the area goes up fourfold and the rim only doubles, while the eccentricity stays exactly where it was. An ellipse scaled up is the same shape, and the number that says what shape it is has no reason to move.
I entered the longer measurement in the shorter box. Why an error instead of a swap?
Because swapping would put a figure on screen that disagrees with the label above the box it came from, and you would have no way of telling that it had happened. The page tells you which box the longer measurement belongs in and leaves the decision with you — it is one edit away from being right, and it stays your measurement rather than becoming the page's.
Is this the same as the circle area page?
For an ellipse whose two axes are equal it agrees with it exactly, and the first row of the table is that case: a 10 by 10 ellipse is a circle of radius 5, and both pages print 78.5398 for the area and 31.4159 for the distance around. For any other ellipse the circle page does not apply at all, because a circle has only one measurement and an ellipse has two.

References

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