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### Radius from an Arc and a Chord

```
Date: 06/08/99 at 19:38:11
From: Jackie Castellano
Subject: Geometry, arcs

If I know the height of an arc from the midpoint on a chord, and the
length of the chord, can I find the radius of the circle of which the
arc is a part? I can't find a formula.
```

```
Date: 06/09/99 at 12:11:16
From: Doctor Peterson
Subject: Re: Geometry, arcs

Hi, Jackie.

You could work out this formula from what is in our FAQ on Circle
Formulas:

http://mathforum.org/dr.math/faq/formulas/faq.circle.html

Look down the page for segments of circles, namely

c = ... = 2 sqrt[h(2r-h)]

You're given h and c, and need to find r, so you can solve this for r
as

r = (c^2/(4h) + h)/2

c^2 + 4 h^2
= -----------
8h

You can work out the formula this way:

***********
*****           *****
****                  /| ****
**                     / |     **
*                      /  |       *
**                    r /   |        **
*                       /    |c/2       *
*                       /     |           *
*                      /      |           *
*                      /  r-h  |      h     *
*                     *--------+------------*
*                      \       |            *
*                      \      |           *
*                       \     |           *
*                       \    |          *
**                      \   |        **
*                      \  |       *
**                     \ |     **
****                  \| ****
*****           *****
***********

We have a right triangle, and Pythagoras gives:

r^2 = (r-h)^2 + (c/2)^2

r^2 = r^2 - 2rh + h^2 + c^2/4

2rh = h^2 + c^2/4

h^2 + c^2/4   4h^2 + c^2
r = ----------- = ----------
2h            8h

I hope that meets your needs.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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