GeminiHeat wrote:
Approximately what percent of the area of the circle shown is shaded, if polygon ABCDEF is a regular hexagon?
(A) 24%
(B) 30%
(C) 36%
(D) 42%
(E) 48%
Attachment:
The attachment polygon.png is no longer available
Let each side of hexagon be 6
Here, AEC is an equilateral triangle with each side \(6\sqrt{3}\)
Area of shaded region = Area of hexagon - Area of equilateral triangle
= \(6\frac{\sqrt{3}}{4}6^2 - \frac{\sqrt{3}}{4}(6\sqrt{3})^2\)
= \(3(\frac{\sqrt{3}}{4}6^2)\)
Now, what percent of the area of the
circle shown is shaded = \(\frac{Ar. of shaded region}{Ar. of circle}\)
= \(\frac{3\sqrt{3}}{4π} = 41.34\)%
Hence, option D
Key points:1. AEC is an equilateral triangle
2. Radius of circle is equal to the side of regular hexagon
3. A hexagon's area equals the area of 6 equilateral triangles with same side length
Attachments
Eq. triangle in Hexagon.jpg [ 599.09 KiB | Viewed 1051 times ]