Associated Topics || Dr. Math Home || Search Dr. Math

### Circle Radius from Chord Length and Depth

```
Date: 06/16/99 at 02:18:52
From: Jacques Duranleau
Subject: Finding Radius of circle knowing only chord and depth.

How do you find the radius or diameter of a circle when you only know
a chord length and the depth? In this case, the chord does not pass
through the center.
```

```
Date: 06/16/99 at 08:45:46
From: Doctor Peterson
Subject: Re: Finding Radius of circle knowing only chord and depth.

Hi, Jacques.

You can almost find this formula in our FAQ on circles. On this page,
scroll down to Segment of a Circle:

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

Unfortunately, it's not quite there, though you could work it out from
what is there, namely

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

You have h (the height of the chord) and c (the length of the chord),
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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search