Thread: Area of Any Triangle

Is there a way to find the area of a triangle by only knowing the lengths of each side? I have been trying to find a way (and am almost done) I would just like to know if this has been solved before...

Is this what you mean?
Heron's Formula -- from Wolfram MathWorld

Originally Posted by Plato

Is this what you mean?
Heron's Formula -- from Wolfram MathWorld

Yes, I've never really seen that before

Hello, Quick!

Is there a way to find the area of a triangle by only knowing the lengths of each side?
I have been trying to find a way (and am almost done).
I would just like to know if this has been solved before.

Yes . . . a long time ago . . .


Heron's Formula

If a, b, c are the sides of the triangle and s = (a + b + c)/2, the semiperimeter,
. . . . . . . . . . . . . . . . . . . . . . . . . _________________
then the area is given by: . A . = . √s(s - a)(s - b)(s - c)

I can post a proof! (The one I knew from 8th grade, still remember)
(Once LaTeX is installed)

Originally Posted by Quick

Is there a way to find the area of a triangle by only knowing the lengths of each side? I have been trying to find a way (and am almost done) I would just like to know if this has been solved before...

Another longer way would be to find one of the angles using the cosine rule and then use that angle and the lengths of the sides that make that angle to find the area.

Originally Posted by Glaysher

Another longer way would be to find one of the angles using the cosine rule and then use that angle and the lengths of the sides that make that angle to find the area.

Well I was trying that and then I thought, "does this work for non right angle triangles"

Originally Posted by Quick

Well I was trying that and then I thought, "does this work for non right angle triangles"

Yes, both the Law of Sines and Cosines works for any triangle.

-Dan

Originally Posted by topsquark

Yes, both the Law of Sines and Cosines works for any triangle.

-Dan

What are the law of Sines and Cosines?

Originally Posted by Quick

What are the law of Sines and Cosines?

Given a triangle with sides a, b, and c and angles A, B, and C across from that side (respectively) (ie. angle A is across from side a, etc.)

Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines:
a^2 = b^2 + c^2 - 2bc*cos(A)
b^2 = a^2 + c^2 - 2ac*cos(B)
c^2 = a^2 + b^2 - 2ab*cos(C)
(Notice the pattern.)
(Note also that the Law of Cosines is a generalization of the Pythagorean Theorem to a triangle that is not a right triangle.)

-Dan

Originally Posted by topsquark

Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)

It it even better to use the "generalized law of sines" which states:
a/sin(A) = b/sin(B) = c/sin(C)=2R
Where R is the radius of the circle circumscribing the triangle.