# How do I find the necessary arc length if I want a certain height for an arch in building?

I am trying to build a hoop house/greenhouse. I will need to bend my poles so that there is enough height but still wide enough at the ground. I cannot figure out how to calculate it.

If I have a 14' wide base, that's the diameter of the circle, radius 7. I also thought the 7 here was a chord for the formulas.

I tried this formula, Sagitta = 7 - sqrt(7^2-7^2). This gives me 7. But I don't know how that helps me because I need a piece of PVC that will give me that Sagitta height, and I do not "see," and maybe this is where I am wrong, that a 14' piece of PVC will bend enough to give me 7' high, if the ground is 14'.

What am I doing wrong (or right)? I thought this would give me the length of the arc necessary and I'm not real sure of the angle I would bend it at to get that arc.

I have looked here, How do I calculate the height of an arc?

radius = distance? arc length = height?

and I have read various things on calculating the Sagitta, like from here:

https://www.mathopenref.com/sagitta.html

This one might be what I need? I admit I didn't understand how to do the math.

In a circle, is there a formula for the length of the sagitta if the chord length and arc length are known?

• What is your ideal height for the hoop house?
– an4s
Sep 4, 2020 at 19:23
• This tool may be of help to you.
– an4s
Sep 4, 2020 at 19:29
• @an4s I picked 7' but it should probably be about 8-10. Thanks for the tool. Edit: 500 Internal Error. Will try again later. Sep 4, 2020 at 19:55
• Ah, sorry, I missed part of the URL. Try this: handymath.com/cgi-bin/arc18.cgi?submit=Entry
– an4s
Sep 4, 2020 at 20:04

If your hoop house is to be $$14$$ feet wide and $$7$$ feet high at the center then it will be a semicircle. Your PVC must be as long as half the circumference of a circle of radius $$7$$, so $$7\pi \approx 22 \text{ feet}.$$
Edit: For an arbitrary height of more than $$7$$ feet you can't have a circular arc.