# [SOLVED] semi-circle

• Jun 20th 2006, 07:17 AM
dnoster
[SOLVED] semi-circle
hi everyone
please could someone help me with a problem. If i have a semi-circle, and at the baseline , at certain intervals on that baseline, i need to know what the height is to the arc of the semi-circle , at right angles to the baseline , would anyone have a formula i could use

cheers
• Jun 20th 2006, 08:23 AM
Soroban
Hello, dnoster!

Quote:

I have a semicircle, and at certain intervals on the diameter, I have points.
I need to know the height from the points to the arc of the semicircle.
Code:

          C           * * *         */: \    *       * / :  \    *     * /  :    \    *       /  :h      \     */    :        \ *     *-----*--+--------*     A  x  P  2r-x    B
A triangle inscribed in a semicircle is a right triangle: .$\displaystyle \angle C = 90^o$

We have a point $\displaystyle P$ on the diameter, where $\displaystyle AP = x.$

If $\displaystyle r$ is the radius of the circle, then $\displaystyle PB = 2r - x$

. . . . . . . . . . . . . . . . . Theorem
. . . . . The altitude to the hypotenuse of a right triangle
. . . . . . . . . . . . .is the mean proportional
. . . . . . . . of the two segments of the hypotenuse.

That is: .$\displaystyle h^2 \:=\:x(2r-x)$

Therefore: .$\displaystyle h\:=\:\sqrt{x(2r-x)}$
• Jun 20th 2006, 10:47 AM
earboth
Quote:

Originally Posted by dnoster
hi everyone
please could someone help me with a problem. If i have a semi-circle, and at the baseline , at certain intervals on that baseline, i need to know what the height is to the arc of the semi-circle , at right angles to the baseline , would anyone have a formula i could use cheers

Hello,

here is another way to do your problem:
Let the center of your circle be the origin of a coordinate system. Then you can use the Pythagorian theorem:

x-value = b
height to the arc of the semi-circle = a

Then you get:

$\displaystyle a^2+b^2=r^2\ \Longrightarrow \ a=\sqrt{r^2-b^2}$

I've attached a diagram to show what I've calculated.

Greetings

EB