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# Determining an Oval's Perimeter Measurement

## By Joan Whetzel

In geometry, we study points, lines, angles, surfaces, shapes (i.e. circles and ovals) and solids (e.g. a sphere, or elliptical sphere). We want to know their properties, their relationship to each other and how to measure them. Take a drawing of an oval, for instance. It has no angles to examine or measure. It doesn't have points around its surface like an egg would. It does, however, have a line defining its outline called the perimeter.

The length of the oval's perimeter can be determined through the use of calculus. But that's too complicated. A rough estimate can be obtained by taking two measurements inside the oval - the semi-major axis and the semi-minor axis - and performing this geometry equation:

2pi x square root [1/2 (a^{2} x b^{2})]

where "a" is the semi-major axis (half of the long diameter, or the long radius) and "b" is the semi-minor axis (half of the short diameter, or the short radius). All you will need to perform this geometry equation is an oval diagram, a ruler, pencil and paper and a calculator with the square root function.

**The Semi-Major Axis and the Semi-Minor Axis **

Circles have only one diameter measurement and radius measurement because any line that bisects a circle will have the same measurement in all directions. An oval, on the other hand has a major axis (diameter that runs the long way across the oval) and a minor axis (diameter that runs the short way across the oval).

Place the diagram so that the oval lays horizontally, pencil and ruler and draw a horizontal line forming the major axis and a vertical line forming the minor axis. Place a small dot at the point in the center where these lines intersect - the vertex. Use the ruler to measure the semi-major axis from the vertex to the perimeter. Do the same for the semi-minor axis. On a separate sheet of paper, write down your measurements.

Let's say you found that the semi-major axis equaled 9 inches and the semi-minor axis came to 6 inches. Then write a = 9 inches and b = 6 inches.

**Solving the Parenthetical Equation **

Now let's look at that equation again -- *2 pi x square root [1/2 (a ^{2 }+ b^{2})]* -- and figure out the back half of it. Begin by plugging in your measurements for the "a" and "b". It should read: [1/2 (9

^{2}+ 6

^{2})].

Begin by squaring both numbers, and then adding them together: 9 squared is 81 and 6 squared equals 36. So 81 + 36 = 117. Now, 117 needs to be multiplied by one-half (0.5), so pull out your calculator and multiply -- 117 x 1/2 (0.5) = 58.5. Make sure you're writing down the equation and results as you go: 2 pi x square root [58.5]. Great! Now it's time to move onto the front half of the equation.

**Finding the Square Root **

The next part of the equation involves finding the square root of the result we just found. Using your calculator, punch in the result - 58.5 - the "x" key and then the square root(√) key, and write down the answer. It should read 7.6485293. Round this up to two places behind the decimal point (7.65) to make it easier. So now the equation looks like this: 2 pi x 7.65

**The Pi part of the equation**

First, multiply pi (π, or 3.14159) by 2 which equals 6.28318. Next multiply it by the square root result we just found: 6.2318 x 7.65 = 48.06. So we have found that the perimeter length for our oval is 48.06 inches in length.