Expert: Sherry Wallin - 5/31/2010

Question
A surveyor determines that the distances from a house to each of the endpoints of the diameter of a small circular lake are 1000m and 2500m.  If the angle between these two sides of the resulting triangle is 72 degrees, what is the area of the lake?

Answer
Ann~
    If I assume that the house is on the edge of the lake then that gives me a triangle with sides 1000m and 2500m and the third side the diameter of the circle x meters. Using the law of cosines and letting the side that is x meters be 'a' I have a^2 = b^2 + c^2 - 2bc*CosA
= 1000^2 + 2500^2 - 2*1000*2500*cos 72 deg which gives me a^2 = 5704915 so a ~= 2388.5 and the area of a circle is pi*r^2. Our 'a' here is the diameter of a circle so we have pi*(2388.5/2)
~= 4,480,630 m^2. Feel free to follow this up with questions if this is not clear.

Math Prof