How do I figure length of a tube forming an arc, knowing only the width of the base & the height of arc?

Question

I have a planter & am bending a PVC pipe in an arc to put a frost cover on. The base of the planter where the 2 ends of the pipe connect to the ground on either side is 3 feet wide, and I want the arc (a sharper angle than a half circle) to reach a height of 4 feet. I have a couple other planters with different widths. What formula do I put in my calculator to get an answer saying how long the pipe should be, given whichever width & height I'm working with?

Answer 1

Assuming you bend the pipe into the shape of half an ellipse, there is an approximation formula for the perimeter of an ellipse that you could use.

The general equation for an ellipse is

$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

and the approximation for the perimeter is

$$P\approx2\pi\sqrt{\frac{a^2+b^2}{2}}\tag{1}$$

For your application, $a$ would be half the width $W$ and $b$ would be the height $H$. And the length $L$ of the pipe would be only half the perimeter of the ellipse. Putting all this together using equation $(1)$ would give

$$ L=\frac{P}{2}\approx\pi\sqrt{\frac{\left(\frac{W}{2}\right)^2+H^2}{2}} $$

In the case where $H=4$ and $W=4$ this gives $L\approx\pi\sqrt{10}\approx9.93$ ft.

ADDENDUM: Follow this LINK to an adjustable pattern you can use to create a form for bending the pipe. You can change the values of $W$ and $H$. I assume you know how to heat the pipe so it will hold its shape. Using the adjustable diagram you can use a sheet of $4\times8$ plywood and tack a pair of finishing nails at the appropriate distances along the angular grid to create a frame for bending each half of the ellipse. Bend one half and after it has cooled, use the same frame for bending and heating the other half. I am attaching a picture of what the link looks like when $H=W=4.$ At the link above you can change those values.