# Figure the Dimensions of an Ellipse Given the Area and Perimeter

The area and perimeter of an ellipse depend on the width and length. However, if you know only the area and perimeter, you can use numerical methods to solve for the width and length. In the case of a circle, simple algebra will suffice to find the diameter. If A and P are the area and perimeter respectively and D is the diameter, then

D = sqrt(4A/π), and

D = P/π

In the case of a circle, P² = 4πA. Among all ellipses of equal perimeter, a circle has the greatest area. Therefore, for any non-circular ellipse, we have the inequalities

A < P²/(4π), and

P > 2⋅sqrt(πA)

The calculator on the left will solve for the half-major and half-minor axes and width and length of an ellipse with a given perimeter and area. In the case of a circular ellipse, it will output the diameter and radius. You can input values with π and square roots.

### Example 1

Bill wants to build a ellipse-shaped table that has an area of about 20 square feet and a perimeter of about 17 feet. How long and how wide should the table be to match these specifications?If A = 20 and P = 17, then P is greater than 2⋅sqrt(πA), so these are acceptable area and perimeter measurements. Using the calculator above, we obtain

Width = 3.7004 feet

Length = 6.8816 feet

### Example 2

An elliptical pool has a uniform depth of 1 meter and a capacity of 90,430 liters. Its perimeter is 34.68 meters. What are the width and length of the pool?Since 90,430 liters equals 90.43 cubic meters and the height of the pool is 1 meter, the surface area is 90.43 square meters. Thus, A = 90.43 and P = 34.68. Using the calculator gives us a length of 13.05 meters and a width of 8.82 meters.

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