Elementary Math – Mensuration – Area of Sector in terms of r

Step 1: Find the Arc Length (s) in terms of r.

$\text{Arc}\ \text{length}\ \text{(s)}\ \text{=}\ \text{200}\ \text{-}\ \text{2r}$

Step 2: Find the Angle (θ) in terms of r.

$\displaystyle \text{Formula: Angle}\ \text{ }\!\!\theta\!\!\text{ }\ \ \text{=}\frac{{\text{Arc}\ \text{Length}\ \text{(s)}}}{{\text{Radius}\ \text{(r)}}}$

$\text{Angle}\ \text{( }\!\!\theta\!\!\text{ )}\ \text{=}\ \frac{{\text{200}\ \text{-}\ \text{2r}}}{\text{r}}$

Step 3: Find the Area of Sector in terms of r.

$\displaystyle \text{Formula: Area}\ \text{of}\ \text{Sector}\ \text{=}\ \frac{\text{1}}{\text{2}}\text{ }\!\!\times\!\!\text{ radiu}{{\text{s}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ Angle ( }\!\!\theta\!\!\text{ )}$

$\displaystyle \begin{array}{l}\text{Area}\ \text{of}\ \text{Sector}\ \\\text{=}\ \frac{\text{1}}{\text{2}}\text{ }\!\!\times\!\!\text{ }{{\text{r}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }\frac{{\text{200}\ \text{-}\ \text{2r}}}{\text{r}}\end{array}$

$\displaystyle \frac{\text{1}}{\text{2}}\text{ }\!\!\times\!\!\text{ }{{\text{r}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }\frac{{\text{200}\ \text{-}\ \text{2r}}}{\text{r}}$

$\displaystyle \frac{{\text{r(200}\ \text{-}\ \text{2r)}}}{\text{2}}$

$\displaystyle \frac{{\text{2(100r}\ \text{-}\ {{\text{r}}^{2}}\text{)}}}{\text{2}}$

$\displaystyle \text{100r}\ \text{-}\ {{\text{r}}^{2}}$

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