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Square Barn - Step 3

Let's break out the triangle separately so we can get a good look at it.

Since the final triangle in this case is isoceles my initial approach was to calculate the altitude and solve by Pythagoras, but later published problems resulted in final scalene triangles where the altitude was not as easily found. I realized that I did not even need the altitude. Everything is determined by the sides, which we know. Here's how to do it:

Find angles by the  Law of Cosines
A= acos((b2+c2- a2)/(2bc))
A
= acos((14.142+402-402)/(14.14*40*2))
A=79.82

 With Angle A known, then Area K = b*c*sinA/2
K = 40*14.14*.984/2 =278.39
or Heron's Rule: K= sqrt(s(s-a)(s-b)(s-c))
where s= (a+b+c)/2

Quadrilateral:
Triangle - 1/2 barn
=278.39 - 50 = 228.39

Angle of wedge = expected rotation of radius to reach base of triangle minus Angle A
=135 - 79.82 = 55.18

so total grazing area = 3/4 of a 50 ft. circle
plus 2 x 55.18/360 of a 40 foot circle
plus quadrilateral 228.39

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