# circles touching internally/area...

• Feb 21st 2010, 07:42 AM
snigdha
circles touching internally/area...
Two circles touch internally. The sum of their areas is 116π sq. cm and distance between their centres is 6cm. Find the radii of the circles.
• Feb 21st 2010, 09:22 AM
Diagonal
Quote:

Originally Posted by snigdha
Two circles touch internally. The sum of their areas is 116π sq. cm and distance between their centres is 6cm. Find the radii of the circles.

Two simultaneous equations:

R - r = 6

πR^2 - πr^2 = 116π
EDIT: That should be R^2 + r^2 = 116.

Sorry if that error misled you.
• Feb 21st 2010, 09:45 AM
Soroban
Hello, snigdha!

Another approach . . .

Quote:

Two circles touch internally.
The sum of their areas is 116π cm².
The distance between their centres is 6 cm.
Find the radii of the circles.

Code:

              * * *           *          *         *              *       *                *       *                  *     - *        o        * -     : *        |        * :     :          *          6     R  *    *  |  *    *  :     :  *      o      *  -     :    *  *  |  *  *    r     -        * * *        -

The radius of the large circle is $\displaystyle R.$
The radius of the small circle is $\displaystyle r.$
We see that: .$\displaystyle R \:=\:r+6$

The area of the large circle is: .$\displaystyle A_1 \:=\:\pi R^2 \:=\:\pi(r+6)^2$

The area of the small circle is: .$\displaystyle A_2 \:=\:\pi r^2$

The sum of the areas is $\displaystyle 116\pi$

So we have: .$\displaystyle \pi(r+6)^2 + \pi r^2 \:=\:116\pi$

. . Now solve for $\displaystyle r.$
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