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The dart board shown is divided into twenty equal sectors. If the diameter of the board is 18 inches, what area of the board does each sector cover?


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If the board is divided into 20 equal sectors, then the central angle of each sector has a measure of 360°÷20=18°360\text{\textdegree} \div 20=18\text{\textdegree}. So, the measure of a sector's central angle is 18°18\text{\textdegree} and the radius is 18÷2=918\div 2=9 inches.

We convert the central angle measure to radians.

18°=18°(πradians180°)=π10\begin{align*} 18\text{\textdegree}&=18\text{\textdegree}(\dfrac{\pi \quad radians}{180\text{\textdegree}}) \\ &=\dfrac{\pi}{10} \end{align*}

We use the central angle and the radius to find the area of a sector.

A=12r2θ=12(9)2(π10)12.7\begin{align*} A&=\dfrac{1}{2}r^2\theta \\ &=\dfrac{1}{2}(9)^2(\dfrac{\pi}{10}) \\ &\approx12.7 \end{align*}

Therefore, each sector covers an area of about 12.7 square inches.

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