If the lengths of the altitudes of a triangle are 4, 5, and 6, what is the area of the triangle?

Can you generalize?

From the Dr. Math Forum , the area of a triangle equals

 1/sqrt[(1/ha + 1/hb + 1/hc)*(-1/ha + 1/hb + 1/hc)*
             (1/ha - 1/hb + 1/hc)*(1/ha + 1/hb - 1/hc) ],
             
where ha, hb and hc are the altitudes of the triangle. That web site derives this formula from Heron's formula.

Working this out for altitudes of 4, 5 and 6 results in

sqrt(2^8*3^4*5^4/(7*13*17*37))

= 3600 / sqrt(57239)

 

Posted by Charlie on 2007-01-15 23:21:07