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Math Help - Area of Any Triangle

  1. #1
    MHF Contributor Quick's Avatar
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    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...
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by Plato View Post
    Yes, I've never really seen that before
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    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)

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    I can post a proof! (The one I knew from 8th grade, still remember)
    (Once LaTeX is installed)
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    Member Glaysher's Avatar
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    Quote Originally Posted by Quick View Post
    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.
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by Glaysher View Post
    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"
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Quick View Post
    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
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by topsquark View Post
    Yes, both the Law of Sines and Cosines works for any triangle.

    -Dan
    What are the law of Sines and Cosines?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Quick View Post
    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
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    Quote Originally Posted by topsquark View Post

    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.
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