Transcription
This is widely considered one of the hardest math problems ever put on a test. You're given a quarter circle with a radius of six, and there's a rectangle sitting inside of it. The width plus the height of this rectangle is exactly eight. So, what is the exact perimeter of the shaded region?
Let's break down the boundary of this complex shape piece by piece. First, we have this large curved arc on the outside. It's exactly 1/4 of a circle. The circumference of a full circle is 2 pi times the radius. So, the full circle is 12 pi. That means our 1/4 arc is exactly 3 pi.
Now, let's look at the two straight flat pieces on the left and bottom edges. We know the total radius of this circle is six. So, the left piece is exactly six minus the height of the rectangle. And the bottom piece is exactly six minus the width of the rectangle. If we add these two straight-edge pieces together, we get 12 minus the height minus the width. But, look back at the problem. It tells us that the height plus the width of the rectangle is exactly eight. So, 12 minus eight means the combined length of those two flat edges is exactly four.
Now, we have the curved arc and two flat edges. There's only one boundary left. The diagonal line cutting straight through the middle. This is where 99% of people fail. How do you find the length of this diagonal line when you don't know the sides of the rectangle? Well, you don't need to. Look at this shape. It's a perfect rectangle. In any perfect rectangle, the two diagonals are always the exact same length. So, if we visually draw the other diagonal starting from the center of the circle and going to the opposite corner, what is it? It touches the edge of the circle. It's literally just the radius of the circle. So, the diagonal is exactly six.
Now, we just add our boundaries together. 3 pi plus 4 plus 6. The perimeter of the shaded region is exactly 10 + 3 pi.