Simple differentiation question

[semc]

A Norman window has the shape of a square surmounted by a semicircle.
The base of the window is measured as having 80cm with a
possible error in measurement of 0.1cm. Use differentials to estimate
the maximum possible error in computing the area of the window.

So what i did was Area=4r²+0.5$$\pi$$r² differentiated wrt r and sub in value of r=40 and dr=0.1. However i got 32+4$$\pi$$. If i were to use Area=r²+0.5$$\pi$$(r/2)² and repeat the steps i would get 16 +2$$\pi$$ which is half of my initial answer. So i want to ask why can't i use the initial method? Ain't the two of them the same?

P/S I wanted to write 0.5(Pi)r^2 not 0.5^Pi(r^2) can't seem to change it. Same for the other equation Pi is not raised to the power

 

[clamtrox]

I guess it depends on how you interpret the measurement error. If you say that the base is $$80 \pm 0.1 \mathrm{cm}$$, then the radius should be $$40 \pm 0.05 \mathrm{cm}$$

 

[diazona]

I guess it depends on how you interpret the measurement error. If you say that the base is $$80 \pm 0.1 \mathrm{cm}$$, then the radius should be $$40 \pm 0.05 \mathrm{cm}$$

Well it's not really a question of interpretation, but that's right... if you take into account the fact that the measurement error in the radius is half of the measurement error in the full side length, then you get the same answer no matter which formula you use.