Multiple choice

A regular hexagon is inscribed in a circle of radius r By how much is the area of the circle is more than the area of the hexagon?

  1. $\displaystyle \left ( \pi -2\sqrt{3} \right )r^{2}$
  2. $\displaystyle \left ( \pi-\sqrt{3} \right )r^{2}$
  3. $\displaystyle \left ( \pi -\frac{\sqrt{3}}{2} \right )r^{2}$
  4. $\displaystyle \left ( \pi -\frac{3\sqrt{3}}{2} \right )r^{2}$
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D Correct answer
AI explanation

A regular hexagon consists of 6 equilateral triangles with side length r, so its total area is 6 times (the square root of 3 divided by 4) times r squared, which simplifies to (3 times the square root of 3 divided by 2) times r squared. The area of the circumscribed circle is pi times r squared. The difference between the circle area and the hexagon area is (pi minus 3 times the square root of 3 divided by 2) times r squared.

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