Transcription
This is one of the hardest geometry questions on the SAT. They give you a circle and they inscribe a hexagon inside of it. The radius is 10 and what you have to do is find the area of those shaded slivers you see on the outside.
It's difficult to know where to start with this. Is there a hexagon formula I should have remembered? Should we be using sine and cosine?
Think about a regular hexagon. That's kind of just six perfect equilateral triangles snapped together. Because the radius of the circle is 10, then every single side of those triangles is exactly 10, which is a nice round number.
Now, let's just drag and drop our formulas. The area of an equilateral triangle is side squared times root 3 over 4. 10 squared is 100. 100 over 4 is 25. So, one triangle is 25 root 3. But, we have six of them. 6 * 25 is 150. So, the entire hexagon is exactly 150 root 3.
Now, look at the full circle. The area is pi r squared. So, that's 100 pi. To find the shaded slices, we just take the whole circle and carve out the hexagon. 100 pi minus 150 root 3. The answer C.