Calculate the Area of an Ellipse

An ellipse is a two-dimensional shape that looks like a flat circle. The area equation for an ellipse will look familiar if you've studied circles before. The key point to remember is that an ellipse has two important lengths to measure: the major radius and the minor radius.

Steps

Calculating the Area

  1. Find the major radius of the ellipse. This is the distance from the center of the ellipse to the farthest edge of the ellipse. Think of this as the radius of the "fat" part of the ellipse. Measure it or find it labeled in your diagram. We'll call this value a.
    • You can call this the "semi-major axis" instead.[1]
  2. Find the minor radius. As you might have guessed, the minor radius measures the distance from the center to the closest point on the edge. Call this measurement b.
    • This is at a 90º right angle to the major radius, but you don't need to measure any angles to solve this problem.
    • You can call this the "semi-minor axis."
  3. Multiply by pi. The area of the ellipse is a x b x π. Since you're multiplying two units of length together, your answer will be in units squared.
    • For example, if an ellipse has a major radius of 5 units and a minor radius of 3 units, the area of the ellipse is 3 x 5 x π, or about 47 square units.
    • If you don't have a calculator, or if your calculator doesn't have a π symbol, use "3.14" instead.

Understanding Why it Works

  1. Think of the area of a circle. You might remember that the Calculate-the-Area-of-a-Circle equals πr2, which is the same as π x r x r. What if we tried to find the area of a circle as though it were an ellipse? We would measure the radius in one direction: r. Measure it at right angles: also r. Plug it into the ellipse area formula: π x r x r! As it turns out, a circle is just a specific type of ellipse.[2]
  2. Picture a circle being squashed. Imagine a circle being squeezed into an ellipse shape. As it's squeezed more and more, one radius gets shorter and the other gets longer. The area stays the same, since nothing's leaving the circle.[3] As long as we use both radii in our equation, the "squashing" and the "flattening" will cancel each other out, and we'll still have the right answer.

Tips

  • If you want a rigorous proof, you'll need to learn how to integrate, a calculus operation.[4]

Related Articles

  • Calculate the Area of a Sector

Sources and Citations

  1. ↑ http://www.mathopenref.com/ellipsesemiaxes.html
  2. ↑ http://www.mathopenref.com/coordgeneralellipse.html
  3. ↑ Su, Francis E., et al. "Area of an Ellipse." Math Fun Facts. https://www.math.hmc.edu/funfacts/ffiles/10006.3.shtml. Accessed August 9, 2015.
  4. ↑ http://www.edmath.org/MATtours/ellipses/ellipses1.10.3.html
Retrieved from "https://kipkis.com/index.php?title=Calculate_the_Area_of_an_Ellipse&oldid=308744"