Angle, radius and the area between a circle and an inscribed regular hexagon from an arc length

Angle, radius and the area between a circle and an inscribed regular hexagon from an arc length

6 marks

A regular hexagon is drawn inside a circle with centre \(O\) and radius \(r\text{ cm}\), so that all six vertices of the hexagon lie on the circle. Two neighbouring vertices are \(A\) and \(B\), and \(\theta\) is the size of angle \(AOB\) in radians. The minor arc \(AB\) has length \(7\text{ cm}\). The six regions inside the circle but outside the hexagon are shaded.

  1. Find
    1. \(\theta\)
    2. \(r\).[3]
  2. Find the total area of the shaded regions.[3]

Solution:

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Answer:(a)(i) \(\theta=\dfrac{\pi}{3}\) (a)(ii) \(r=\dfrac{21}{\pi}\approx6.68\text{ cm}\) (b) \(24.3\text{ cm}^2\)

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