Circle To Annulus

Geometry Level 1

Given rectangle ABCD, we draw 3 concentric circles centered at A with radii AB, AD, and AC, respectively.

Which has a larger area, the green circle or the yellow annulus?

Green circle Yellow annulus They are equal Depends on the rectangle

1 solution

Geoff Pilling
Oct 28, 2016

Relevant wiki: Length and Area Problem Solving

Lets call the radii, in increasing order, $r_1$ , $r_2$ , and $r_3$ . Now, if we consider the diagonal of the rectangle (which is $r_3$ ), then the Pythagorean theorem says:

$r_1^2 + r_2^2 = r_3^2$

$r_1^2 = r_3^2 - r_2^2$

And the area of the smallest circle is $\pi r_1^2$

And the area of the yellow region is $\pi (r_3^2 - r_2^2)$

Therefore, $\boxed{\text{they are equal}}$ .

I found it somewhat amazing that these two areas are identical! Such a nice property hidden in the rectangles and circles.

Calvin Lin Staff - 4 years, 7 months ago

Circle To Annulus

1 solution

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