Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)?

108 – 18
54 rt3 – 9
54 rt3– 18
108 – 27
54 rt3 – 27





If I use the Pitag. theo. to solve it instead of the 30-60-90 rule, I get a trianbgle that is 6 of hypot. and 6/2 that would give me a third side of 5, right?
Am I missing something?

Thanks

Ruben

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I'm not sure exactly what you're trying to say here, but: If you draw a right triangle (by erecting a side from the center, perpendicular to one of the sides of the hexagon), then the short leg of the triangle is 6/2 = 3, and the hypotenuse is 6. So the third side is root(6^2 - 3^2), which is root(36 - 9) = root(27) = 3*root(3).

What I think is happening here is that you're subtracting 36 - 9 and getting 25. It's a common mistake, right along with the likes of 16 - 9 = 5, and other such digit-switching errors.

By the way, I'm quite impressed with your ability to embed images within posts.

Thank you so much!

I guess when you study too long even a easy subtraction becomes an obstacle;)

got it. totally get the explanation provided and how to solve this question as explained in the CAT.

However, can you please show me the error of my ways and tell me what is wrong with calculating the area of the hexagon using two trapezoids (resulting in the incorrect answer, D,) rather than six equalitaral triangles? This is more for future problems going forward so I don't actually make this error on the exam.

Thanks!

my mind-reading is a little bit off today, so, unfortunately, i can't tell you what's wrong with your solution unless you post it.

using two trapezoids should still give you the correct answer of 54√3 for the hexagon's area, though:
each trapezoid has height 3√3 (see my post above - this is almost certainly where you made your mistake)
each trapezoid has base1 = 6 and base2 = 12
so area of each trapezoid = (1/2)(3√3)(6 + 12) = 27√3
so hexagon = 2(27√3) = 54√3