Ellipse Area Calculator

This ellipse area calculator allows you to determine the area of an ellipse from its two axes or vice versa. Enter any two values (major radius, minor radius, area) and leave the third field blank, then click "Calculate" to find the missing value.

The units are automatically converted during the calculation so that you can mix centimeters with inches for example and get the area in square feet. Besides the area, this calculator also calculates the circumference, eccentricity, and focal length which describe the shape of an ellipse.

What is an ellipse?

An ellipse is an elongated circle, a precisely defined oval shape. It has two axes that are perpendicular to each other; the longer one being the major axis and the shorter one the minor axis. Half of each axis, measured from the center, is the long radius a and short radius b, two values which determine the size and shape of the ellipse.

Within each ellipse there are two special points called foci. Their defining property is very simple: the sum of distances from any point on the curve to the two foci is always equal. This is the principle by which gardeners can draw an ellipse using two stakes and a piece of string. When the two foci coincide at the center, the two axes are of equal length and the ellipse becomes a circle.

Formula for area of an ellipse:

The area of an ellipse is found by multiplying pi by the long radius and then by the short radius.

where A is the area, a is the major axis and b is the minor axis, with π being approximately 3.14159. This formula is an extension of the simplified formula for the area of a circle. A circle is a special case of an ellipse where both axes are equal in length (a = b), each corresponding to the radius r. In this case, the formula reduces to the familiar π*r^2.

Since these three dimensions are related by the same relationship, you can also rearrange the formula to calculate a missing axis instead of area. You divide the area by pi and then by the known axis.

Example:

The answer is calculated as follows for the ellipse area:

Here's how to use this calculator:

Enter any two of the known values and leave the third field blank. To calculate area enter major axis and minor axis. To calculate a missing axis enter area and one known axis.

If you fill two fields, the calculator will calculate and show the third value as a result. Also, it shows perimeter, eccentricity, focal length, and radius of equal area circle. No mode selection is required. Just leave one field empty to tell the calculator what to calculate. If you change unit of any element, other values are automatically converted. By enabling "Volume" option, you can extend a 2D ellipse into a 3D cylinder.

If you use the entire axle instead of just the half axle...

For some problems, instead of the semi-major and semi-minor axes, you'll be given the full lengths of the major and minor axis that cross through the entire ellipse. Since the semi-axis is always half of the full axis, divide each measurement by two before plugging it in. If the major axis is 14 and the minor axis is 6, then the major radius is 7 and the minor radius is 3.

If the total length of the axes is given, a factor of one quarter is added to the area calculation formula. This is because the product of the two axes, each divided by two, equals one fourth of the original value.

Half axles, surface and their variants

The table below shows the same relationship in three different ways, corresponding to the possible missing three magnitudes.

Example calculation

Consider an ellipse with a major radius of 7 centimeters and a minor radius of 4 centimeters. Multiply π by the major and minor radii.

The calculation in the opposite direction is just as easy. Suppose that the area of an elliptical flower bed is 50 square meters and the smaller radius is 3 meters. Then the larger radius is equal to the value obtained by dividing the area by π, and then dividing it by 3, which is approximately 5.31 meters. With this calculation tool you only need to enter two values in order to obtain the third one.

Some examples of area of ellipses:

The following table shows some area values corresponding to various combinations of large and small radii. The results are rounded to two decimal places. As the area is the product of both axes, ellipses with semiaxis lengths of 8 and 2 have the same area as ellipses with semiaxis lengths of 4 and 4, even though they look very different.

Size of an ellipse:

Calculating the circumference of an ellipse has traditionally been a more difficult task than calculating its area. Unlike with a circle, there is no simple and exact formula for the circumference of an ellipse. The actual value is an integral that cannot be expressed in terms of elementary functions. Mathematicians use approximations, and this calculator uses Ramanujan's first approximation formula. For most ellipses, the error is within a smaller percentage.

In the example with a major axis length of 7 and minor axis length of 4 centimeters, the circumference is approximately 35.0 centimeters. The closer the ellipse is to being a circle, the more accurate this approximation will be. In the case of a perfect circle, it will be exactly 2*π*r returned.

Eccentricity and foci

Eccentricity is a value that indicates how elongated an ellipse is. It starts at 0 for a perfect circle and gradually increases to 1 as the ellipse becomes thinner. It corresponds to the ratio between the focal length and the major axis.

Here 'c' represents the distance from the center of an ellipse to one of its foci and is often shown in calculation tools as focal length. For example, a planet's orbit follows an elliptical path with the sun located at one of the two foci. The eccentricity of the orbit indicates how much it deviates from a perfect circular shape. Earth's orbital eccentricity is about 0.017 and thus nearly circular. In contrast, comets can have orbits with eccentricities that exceed 0.9 resulting in very elongated paths.

Circles of equal area

It can be useful to replace an ellipse with a circle of the same area. This is for example when comparing ellipses and circular pipes. When you assume that π*r squared is equal to π*a*b (where a is the major semiaxis and b is the minor semiaxis), it follows that the radius of the corresponding circle is the geometric mean of the major and minor semiaxes.

In the calculator this is represented as the radius of a circle with equal area. For an ellipse with axes lengths of 9 and 4, the radius of a circle with equal area would be 6 since the square root of 36 is 6.

The volume of an ellipsoidal cylinder

If an ellipse is extended vertically in the plane to create a three-dimensional object, it becomes an elliptic cylinder. This shape can be found often in storage tanks, pipelines, ventilation ducts and architectural columns. Its volume is the product of the area of the ellipse and its height.

If you enable the volume option on a calculation tool and enter the height, then the tool will calculate the volume and convert it from cubic centimeters to liters or cubic feet if necessary.

Applications of the ellipse area

Ellipses are found everywhere in science, design and everyday life. Astronomers care about ellipses because planets and comets have elliptical orbits. This law was discovered by Johannes Kepler four centuries ago. Architects and engineers use elliptical arches, domes and whispering galleries. In a whispering gallery, a voice spoken at one focus can be heard clearly at the other focus.

In everyday life, sports tracks often have a shape where two straight lines are connected by elliptical arcs. For elliptical dining tables or carpets, the area needs to be calculated in order to design tablecloths or lay flooring. Engineers determine the cross-sectional dimensions of elliptical pipes and channels to regulate flow. Mechanics cut out elliptical openings, and the allowable flow rate is determined by their area.

Tips for accurate results:

Either measure from the exact center of the ellipse to the curve along each axis, or measure the width of the entire ellipse and then divide by two using a calculator. Make sure your units are consistent, or let the calculator convert them for you. Remember that the major radius is always the longer of the two radii. The order in which you enter the axes does not change the result since area depends only on the two values. Regardless of the order entered, eccentricity and focal length will be based on the longer axis.