SOLUTION: Find the area of a triangle that has sides of length 5, 6, and 7.

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Question 247152: Find the area of a triangle that has sides of length 5, 6, and 7.
Found 2 solutions by Theo, Edwin McCravy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
area of a triangle equals 1/2 * b * h

from your triangle, I don't think we can find the base very easily.

fortunately there is a formula that allows you to find the area of a triangle when you only know the sides.

It's called Heron's formula.

that formula is:

Area of a Triangle = sqrt((s*(s-a)*(s-b)*(s-c)),

where s=(a+b+c)/2 which is the same as p/2 where p equals the perimeter of the triangle.

assuming that c is the base of the triangle, Heron's formula allows us to find the height of the triangle as well.

That formula would be:

h = (2 * (SQRT(s(s-a)(s-b)(s-c))) / 2

I believe this comes from the fact that A = 1/2 * b * h, so if we solve for h, we get h = 2*A/b

In your triangle:

s = (5+6+7)/2 = 18/2 = 9

In your triangle:

A = sqrt(9 * (9-5) * (9-6) * (9-7)) which equals:

sqrt (9 * 4 * 3 * 2) which equals:

sqrt (216) which equals:

14.69693846

Since A = 14.69693846, then h = 2/c * 14.69693846 = 2/7 * 14.69693846 = 4.199125273

If A = 1/2 * b * h, then A = 1/2 * 7 * 4.199125273 = 14.69693846 which is back where we started from.

You can check Heron's formula for yourself by selecting the following link.

http://mste.illinois.edu/dildine/heron/triarea.html

Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!

Hero(n)'s formula states that a triangle that has sides a,b,c, has area:



where 

Just substitute , , and 



Then substitute , , , and  in









, which is about 14.69693846

Edwin
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