help with geometry

The Area of the bottom-most rectangle is 60 * 20 = 800 u².

 

The Area of the "ring" is simply the difference of the two circles with the inner and outer radii.

 

Inner radius = 50 - 20 = 30u

Outer radius = 50u

 

Area = πr_outer² - 30πr_inner²

Area = π50² - π30²

Area = 2500π - 900π 

Area = 1600π u²

 

The Area of the "V" in the center can be simply divided up into two congruent parallelograms and a trapezoid.

 

Area_parrellogram= bh = 30*10 = 300 u²

Since we have 2 of these, the total is 300*2 = 600 u²

 

Area_trapezoid= h(a+b)/2 = 15(10+20)/2 = 15*30/2 = 225 u²

 

Area_"V"= 600 + 225 = 825 u²

 

 

Now the total area is the sum of the areas of the "V", the ring, and the rectangle. (Sum of everything underlined)

 

825 + 1600π + 800 = 1600π + 1625 u²

From above, we can deduce that a = 1600, and b = 1625.

Therefore a+b = 3225 u².

 

I don't mean to criticize, because the work is very good, but an error in the first line messes up the total.  

 

The first line,   The Area of the bottom-most rectangle is 60 * 20 = 800 u².    should equal 1200.