Question #59ecb

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Answer 1 · 2016-09-10T17:19:17

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Let be $s = \frac{L}{n}$ $s = L/n$ where $L$ $L$ is the circumference length and $n$ $n$ the number of slices used.

For each slice we have $b \approx s$ $b approx s$ where $b$ $b$ is the slice (triangle with vertice at the circle's center) basis

Each triangle has an area giving by

$a = \frac{b \times h}{2}$ $a =( bxx h)/2$ where $h = \sqrt{r^{2} - {(\frac{b}{2})}^{2}}$ $h = sqrt(r^2-(b/2)^2)$ (Pythagoras).

but $b \approx s = \frac{L}{n}$ $b approx s = L/n$ so

$A = lim_(n->oo)n(1/2)(L/n) sqrt(r^2-(L/(2n))^2) = L r/2$

but $L = 2 pi r$ so finally

$A = pi r^2$