The formula for calculating the area of a curve in a polar graph is $\large \rm \frac{1}{2}\int r^2 d\theta $ and is adapted from
$\large \rm Area = \frac{1}{2}r^2\times \theta $
But the formula to calculate the arc length is very different from $\large \rm Length =\int r d\theta $ which should've been adapted from $\large \rm Length=r\times \theta $. Why is the formula $\large \rm Length =\int r ~d\theta $ not correct to calculate the arc length of a sector in a polar graph?

Question

The formula for calculating the area of a curve in a polar graph is $\large \rm \frac{1}{2}\int r^2 d\theta $ and is adapted from 
$\large \rm Area = \frac{1}{2}r^2\times \theta $
But the formula to calculate the arc length is very different from $\large \rm Length =\int r d\theta $ which should've been adapted from  $\large \rm Length=r\times \theta $. Why is the formula  $\large \rm Length =\int r ~d\theta $ not correct to calculate the arc length of a sector in a polar graph?

evoker · Answer

I would guess since it does not take into account changes in r.

phi · Answer

The area of a triangle is 0.5* base * height.  The height (altitude) is not the same as the length of a side (e..g. "s" if we are computing arc length)
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