Area of a Triangle

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Heron's Formula

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Key Questions

  • The area of a triangle given two sides and an included angle is given as #A=1/2ab si nC# where a and b are the sides and angle C is the included angle.

    enter image source here

    Example: Given the triangle below, find the area.

    enter image source here
    Solution:
    #A=1/2ab si nC#
    #A=1/2(7)(9) si n30degrees#
    #A=1/2(63)(1/2)#
    #A=63/4# square units

  • Heron's formula allows you to evaluate the area of a triangle knowing the length of its three sides.
    The area #A# of a triangle with sides of lengths #a, b# and #c# is given by:

    #A=sqrt(sp×(sp-a)×(sp-b)×(sp-c))#

    Where #sp# is the semiperimeter:

    #sp=(a+b+c)/2#

    For example; consider the triangle:
    enter image source here
    The area of this triangle is #A=(base×height)/2#
    So: #A=(4×3)/2=6#
    Using Heron's formula:
    #sp=(3+4+5)/2=6#
    And:
    #A=sqrt(6×(6-5)×(6-4)×(6-3))=6#

    The demonstration of Heron's formula can be found in textbooks of geometry or maths or in many websites. If you need it have a look at:
    http://en.m.wikipedia.org/wiki/Heron%27s_formula

  • You can use it whenever you know the lengths of all three sides of a triangle.


    I hope that this was helpful.

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