Question

A triangular lot bounded by three streets has a length of 300 feet on one street, 250 feet on the second, and 420 feet on the third. The smallest angle formed by the streets is 36°. Find the area of the lot.

Answer 1

Apply the Law of Sines.

sin A      sin B     sin C
-------  = -------- = --------
    a            b           c

Inserting the given data, 

 sin 36      sin B     sin C
---------  = -------- = --------
  250          300        420
                                                                   300 sin 36
Then 300 sin 36 = 250 sin B, and sin B = ---------------- = 0.705 
                                                                        250

Thus, arcsin B = 0.783 rad = 44.857 degrees, or about 45 degrees.

Taking B to be 45 degrees and the given angle 36 deg, then the 3rd angle must be 180 - (45+36) = 99 degrees (which is opposite the 420-ft side).

At this point we apply Heron's formula.  You should look this up on the 'Net.

                                      a + b + c        250 + 300 + 420
First, we calculate S = --------------- = ------------------------ = 970/2 = 485
                                            2                       2

and then calculate sqrt (S(S-A)(S-B)(S-C):

Area of triangle / lot is then:

A = sqrt(485(485-250)(485-300)(485-420))
    = sqrt(485(235)(185)(65) )
    = 37021 square feet