Hello, this is problem 19 on the grades 11 and 12, 2025, Math Kangaroo. The area of the black semicircle is 12 centimeters squared. What's the area of the big quarter circle? We have options A, 42 centimeters squared, B, 36 centimeters squared, C, 32 centimeters squared, D, 30 centimeters squared, and E, 25 centimeters squared. Take your time, pause the video. Okay, so it helps to go in and label everything. I'll label my small radius in little r, and I'll call my big radius big R. So then what we can do is we have like some points, and we have some pretty well-defined points. Specifically, what we have are the fact that from here to here is 2r, and that from here down is r, because it is exactly, you know, one radius, and thus this distance here, which is actually equivalent, so this blue line over here is equivalent to big R, can also be written as square root of 2r squared plus r squared, which then means that r squared, big R squared, is equal to 5 little r squared. And then we can compare, yeah, we can compare. So we have that pi times little r squared, all over 2, is equal to 12 centimeters squared. So then if we take the, you know, the area of the big quarter circle, we get that that should be big R squared times pi, all over 4. We can do some substitution to get that, like R squared equals 5, big R squared equals 5 little r squared, so we get that 5 little r squared times pi, all over 4, is equal to, and then you can sort of substitute stuff in, so we can take this 5 over 2 out, times pi r squared, over 2. And we can just substitute this back with the 12 centimeters squared. And then we get that 5 halves of 12 centimeters squared is equivalent to the original area, and that turns out to just be 30 centimeters squared. So our answer is D.

Math Kangaroo 2025

quarter circle area

semicircle area

geometry problem

grades 11 and 12