Finding the area of a triangle, given three sides?

Question

The only way I can think of doing it is graphically, if you have a triangle with sides a, b, c, (c is the longest) and no angles, and you have to find the area, the only way I came up with was drawing a graph and finding out the crossing point of the circles of radius a and b, with centres on the origin and (c,0) respectively, where the x- and y-values are both positive. I then used that y-value as the height and c as the base. But surely there's a better way than that?

How do you prove Heron's formula though? Or at least arrive at it?

sorry i didn't see the proofs at the bottom. thanks.

Answer 1

You can use Heron's formula:

Area=sqrt(s(s-a)(s-b)(s-c)),

where s=(a+b+c)/2 or perimeter/2.

For example, if you had a triangle that was 3-4-5:

s = (3 + 4 + 5) / 2 = 6
Area = sqrt(6 (6-3)(6-4)(6-5) )
Area = sqrt(6 (3)(2)(1))
Area = sqrt(36)
Area = 6 sq. units.

Another example:
a = 3
b = 5
c = 6

s = (3 + 5 + 6)/2 = 8
Area = sqrt(8 * (8-3)(8-5)(8-6))
Area = sqrt(8 * (5)(3)(2))
Area = sqrt(8 * 25)
Area = sqrt(200)
Area = 10sqrt(2) ≈ 14.142 sq. units.

A couple derivations are given below:

Answer 2

Area Of Triangle Given 3 Sides

Answer 3

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RE:
Finding the area of a triangle, given three sides?
The only way I can think of doing it is graphically, if you have a triangle with sides a, b, c, (c is the longest) and no angles, and you have to find the area, the only way I came up with was drawing a graph and finding out the crossing point of the circles of radius a and b, with centres on the...

Answer 4

Here is information regarding Heron's formula...