How do you calculate the length of an arc and the area of a sector?

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How do you calculate the length of an arc and the area of a sector?

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Answer 1 · 2014-12-19T16:33:30

For any $θ$ $theta$ , the length of the arc is given by the formula (if you work in radians, which you should:
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The area of the sector is given by the formula $\frac{θ r^{2}}{2}$ $(theta r^2)/2$

Why is this?
If you remember, the formula for the perimeter of a circle is $2 π r$ $2pir$ .
In radians, a full circle is $2 π$ $2pi$ . So if the angle $θ = 2 π$ $theta = 2pi$ , than the length of the arc (perimeter) = $2 π r$ $2pir$ . If we now replace $2 π$ $2pi$ by $θ$ $theta$ , we get the formula $S = r θ$ $S = rtheta$

If you remember, the formula for the area of a circle is $π r^{2}$ $pir^2$ .
If the angle $θ = 2 π$ $theta = 2pi$ , than the length of the sector is equal to the area of a circle = $π r^{2}$ $pir^2$ . We've said that $θ = 2 π$ $theta = 2pi$ , so that means that $π = \frac{θ}{2}$ $pi = theta/2$ .
If we now replace $π$ $pi$ by $\frac{θ}{2}$ $theta/2$ , we get the formula for the area of a sector: $\frac{θ}{2} r^{2}$ $theta/2r^2$

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How do you calculate the length of an arc and the area of a sector?

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