# Thread: concentric circles

1. ## concentric circles

An archery target consists of three concentric circles. The outer rim of the target has a radius of 1.2m. Find the radii of the two inner circles so that the three areas( central circle and two rings are equal)

2. total area = $\pi(1.2)^2$

inner circle ... $\pi r_1^2 = \frac{\pi(1.2)^2}{3}$

middle ring ... $\pi(r_2^2 - r_1^2) = \frac{\pi(1.2)^2}{3}$

outer ring ... $\pi[(1.2)^2 - r_2^2] = \frac{\pi(1.2)^2}{3}$

3. supposely the answer is 0.4m,0.8m,1.2m

but, that cant be right, eh

4. Hello, requal!

An archery target consists of three concentric circles.
The outer rim of the target has a radius of 1.2m.
Find the radii of the two inner circles so that the three areas
(central circle and two rings) are equal)

Let $a$ = smallest radius.
Let $b$ = next larger radius.
We know that $c = 1.2$ is the largest radius.

The area of the central circle is: . $\pi a^2$ .[1]

The area of the middle ring is: . $\pi b^2 - \pi a^2 \:=\:\pi(b^2-a^2)$ .[2]

The area of the outer ring is: . $\pi(1.2^2) - \pi b^2 \:=\:(1.44 - b^2)\pi$ .[3]

Since the areas are equal, [1] = [2] gives us:

. . $\pi a^2 \:=\:\pi(b^2-a^2) \quad\Rightarrow\quad b^2 - 2a^2 \:=\:0$ .[4]

And [2] = [3] gives us:

. . $\pi(b^2-a^2) \:=\:\pi(1.44 - b^2) \quad\Rightarrow\quad 2b^2 - a^2 \:=\:1.44$ .[5]

Solve the system: . $\begin{array}{cccc}b^2-2a^2 &=& 0 & {\color{blue}[4]} \\ 2b^2 - a^2 &=& 1.44 & {\color{blue}[5]} \end{array}$

. . and we get: . $\boxed{a \:=\:\frac{2\sqrt{3}}{5}} \quad\boxed{ b \:=\:\frac{2\sqrt{6}}{5}}$

5. so, the teacher was wrong after all,
thanks