Area Of Octagon - Problem With Solution

Area Of Octagon - Problem With Solution

Problem:

Find the length of one side, the perimeter and area of a regular octagon given the distance between two opposite sides (span).


circle within right triangle, problem.

Solution to the Problem:

  • The octagon has 8 interior angles. The measure of one interior angle ABD is given by
    (8 - 2)*180 / 8 = 135 degrees

    circle within right triangle, problem.

  • Draw AC and BC so that they are perpendicular at C. Hence the measure of ABC is given by
    135 - 90 = 45 degrees

  • Note that triangle ABC is right and isosceles. We now write that the given distance of 10 cm is the sum of 2y and x

    2y + x = 10

  • Using Pythagora's theorem, we can also write

    2y2 = x2

  • We now solve the above system of equations to obtain x, the length of one side.

    x = 10 / (1 + sqrt(2)) cm

  • The perimeter P is given by.

    P = 8*10 / (1 + sqrt(2)) = 80 / (1 + sqrt(2)) cm

  • The area A may be calculated by subtracting the areas of the 4 right triangles from the area of the large square of side 10 cm.

    A = 10*10 - 4(1/2)(y2) = 100 - 2(y2) = 100 - (x2) = 200(sqrt(2)-1) cm2


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