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Heron's Formula for Area of a Triangle

  • 19-03-2007 11:22pm
    #1
    Precision
    Closed Accounts Posts: 14 ✭


    The usual formula for the area of a triangle as taught in school is,

    Area of Triangle = (1/2)*(Base)*(Perpendicular Height)

    Ok, however there is another formula for the area of a triangle,

    Area of Triangle = (1/4)*SquareRoot((a+b+c)*(-a+b+c)*(a-b+c)*(a+b-c))

    where a, b, c are the lengths of the three sides.
    The formula does not require the perpendicular height.

    This formula is known as Heron's Formula for the Area of a Triangle.

    Example:
    Find the area of a triangle with sides of length 6, 7, 9

    Area = (1/4)*SquareRoot((6+7+9)*(-6+7+9)*(6-7+9)*(6+7-9))

    = (1/4)*SquareRoot((22)*(10)*(8)*(4))

    = (1/4)*SquareRoot(7040)

    = (1/4)*(83.9047)

    = 20.976


Comments

  • #2
    aequinoctium
    Closed Accounts Posts: 667 ✭✭✭aequinoctium


    let s be the semi perimeter of the triangle
    let a, b, c be the lengths of the sides of the triangle

    area = square root[(s)(s-a)(s-b)(s-c)]


  • #3
    Precision
    Closed Accounts Posts: 14 ✭Precision


    The volume of a tetrahedron is,

    Volume of Tetrahedron = (1/3)*(Area of Triangle Base)*(Prependicular Height)

    There is a nice formula for the volume of a tetrahedron as a function of its six edge lengths.

    (Let the faces of the tetrahedron be triangles abc, aef, bdf and ced.
    Therefore, the egdes a and d do not meet, the egdes b and e do not meet, and the edges c and f do not meet.)

    Pease Note... it is not so difficult but it takes a bit of time because are there are a lot of terms to multiply. The terms begin to cancel each other out. So, if time is of the essence coming up to exams or that, don't bother until some other time.

    If not then there is another nice formula that can also be derived...
    The Radius of the Sphere that Circumscribes the Tetrahedron as a function of the tetrahedron's six edge lengths.

    Again, it is not that difficult... just a matter of going through the mathematics... substituting and multipling, etc... and again a lot of the terms cancel each other out.


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