Question 1137781
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Heron's formula LOOKS difficult; but with a little practice it is quite easy to use... although the resulting arithmetic can be rather ugly.<br>
Note you have not stated the formula correctly.  The area is NOT s(s-a)(s-b)(s-c); it is the SQUARE ROOT OF s(s-a)(s-b)(s-c).<br>
Let's use the formula to find the area of this triangular island.<br>
s = (1/2)(a+b+c)<br>
This defines the semiperimeter s of the triangle -- one-half of the sum of the lengths of the three sides.  The first step in using Heron's formula is to determine this number.<br>
In your example, the perimeter is 16+10+8 = 34; so the semiperimeter is 34/2 = 17.<br>
A=sqrt(s(s−a)(s−b)(s−c))<br>
This looks ugly; but, again, it is easy to use, with a little practice.  The formula tells you that, to find the area, you first multiply 4 numbers together: the semiperimeter s, and the semiperimeter minus each of the individual side lengths (s-a), (s-b), and (s-c).<br>
In your problem, s is 17, and the side lengths are 16, 10, and 8.  So the 4 numbers you multiply together are s=17, s-a = 17-16 = 1, s-b = 17-10 = 7, and s-c = 17-8 = 9.  The product is<br>
{{{17*1*7*9}}}<br>
Then the area is the square root of that number:<br>
{{{sqrt(17*7*9) = sqrt(9*119) = sqrt(9)*sqrt(119) = 3*sqrt(119)}}}<br>
Here is another example where the numbers are "nicer": find the area of a triangle with side lengths 6, 8, and 10.<br>
You might recognize this as a right triangle with legs 6 and 8; its area using the familiar formula one-half base times height is (6*8)/2 = 24.  Let's find that result using Heron's formuls.<br>
s = (6+8+10)/2 = 12
s-a = 12-6=6
s-b = 12-8=4
s-c = 12-10=2<br>
sqrt(s(s-a)(s-b)(s-c)) = sqrt(12*6*4*2) = sqrt(12*12*4) = 12*2 = 24<br>