1. ## Precise Elliptical Perimeter

Hello, ellipse perimeter formulas are easily found on google, but I can't find any that are extremely accurate and easily usable.

I found several calculators (and formulas) that came to the same or similar figure, but they are not precise enough.

I measured a 53 x 25 ellipse (very close accuracy), calculated the perimeter (approx 130). I intended to space 60 holes on the line, so I spaced my caliper accordingly, went around and ended up with 62 holes.

Given, even misplacing the caliper a 1/16 of an inch will throw it off, maybe even two holes, I'm quite confident the formula simply wasn't accurate enough.

Formulas?

2. Try 126.5021066. This should be the exact perimeter.

The formula I used:

$\displaystyle 4a\int_{0}^{\frac{\pi}{2}}\sqrt{1-\varepsilon ^2\sin^2\theta}\: d\theta$

where $\displaystyle \varepsilon$ is the eccentricity ($\displaystyle \sqrt{1-\left(\frac{25}{53}\right)^2}$ in your case), and $\displaystyle a$ is the semi-major axis--53/2 in your case.

I doublechecked by plugging in your semi-major and minor axes (a=53/2 and b=25/2 respectively) into Ramanujan's formula:

$\displaystyle C\approx\pi\left(a+b\right)\left(1+\frac{3\left(\f rac{a-b}{a+b}\right)^2}{10+\sqrt{4-3\left(\frac{a-b}{a+b}\right)^2}}\right);\!\,$

which gives 126.502111621. A very close approximation.

When in doubt, and when you don't want to deal with calculus, go with Ramanujan's formula. Except for very flat ellipses it will give you a circumference that is exact to four or five or more decimal places.

I had to look up "caliper" on wikepedia. Are you plotting out a garden?

3. Thank you. I'll try it on my next project; hopefully it will be usable.

I needed the perimeter for cutting gaskets. I needed 60 bolt holes through that pattern. Customers wouldn't be happy if they had 62 LOL

Maybe someday I'll plant a garden though

Thanks again

4. Hmmm, after building an excel calculator with that formula, I realized I had 62 holes based on 130 (2.169 spacing based on 60 hole count) and with that formula/figure at 60 holes I got a 2.108 spacing.

My original spacing needed to be bigger (thus reducing hole count), so in this scenario I'd be reducing the spacing even more.

This formula seems the most precise, but I'm not sure. I'm going to attempt to verify the quality of my tools to ensure they aren't inaccurate.

Edit: It'll be awhile before I can precision tools to compare to mine, but according to my tools & math, the perimeter is approximately 138.75 inches.

5. Caleb-

Go with your instinct. I am a student and am not 100% sure my answer is correct. Hopefully a more authoratative mathematician will stop by and comment on this thread.