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What is the area of an equilateral triangle with height of 9 inches?

    ##A=27 sqrt(3) approx 46.77## inches.

    In such situations, the first step is to draw a picture.

    In relation to the notation introduced by the picture, we know that ##h=9## inches.
    Knowing that the triangle is equilateral makes everything easier: the heights are also medians. So the height ##h## is perpendicular to the side ##AB## and it divides it in two halves, which are ##a/2## long.
    Then, the triangle is divided into two congruent right triangles and the holds for one of these two right triangles: ##a^2=h^2+(a/2)^2##. So ##3/4a^2=h^2## i.e. ##a^2=4/3 h^2##. In the end, we get that the side is given by ##a=[2sqrt(3)]/3 h=[2sqrt(3)]/3 * 9=6 sqrt(3) approx 10.39## inches.
    Now the area:
    ##A=(a*h)/2=([2sqrt(3)]/3 h * h)/2=[sqrt(3)]/3 h^2=[sqrt(3)]/3 81=27 sqrt(3) approx 46.77## inches.

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