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Math Help - Problem Solving #2

  1. #1
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    Problem Solving #2

    Take a circle of radius 5, and mark 12 points that are equidistant around the circle.
    Draw a line segment connecting each of these points with its adjacent points.
    What is the area of the space between the circle and the lines?
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  2. #2
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    Unclear. Diagram?
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  3. #3
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    Hello, matgrl!

    Take a circle of radius 5, and mark 12 points that are equidistant around the circle.
    Draw a line segment connecting each of these points with its adjacent points.
    What is the area of the space between the circle and the lines?

    We have a circle with an inscribed regular dodecagon (12-sided polygon).


    The area of the circle is: . \pi r^2 \:=\:25\pi\text{ units}^2.


    The dodecagon is composed of 12 congruent isosceles triangles
    . . with two sides of length 5 and the included angle 30^o.

    The area of one triangle is: . \frac{1}{2}(5^2)\sin30^o \:=\:\frac{25}{4}\text{ units}^2.

    The area of the dodecagon is: . 12 \times \frac{25}{4} \:=\:75\text{ units}^2.


    Therefore, the difference is: . 25\pi - 75 \;\approx\;3.54\text{ units}^2.
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    Dont have one. This is the only information I was given.
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  5. #5
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    Thank you very much Soroban. I have a few questions that you might be able to clear up for me.

    The area of a triangle is A = 1/2 b * h

    -Why are you then using 5^2?

    Also how did you know to use sin 30 degress =25/4 where did your 4 come from?

    Thank you for your help...this is wonderful!
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  6. #6
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    Do you "see" that 12 isosceles triangles with equal sides = 5 and equal angles = 75 degrees
    are created? If you do, then do you not know how to calculate the area of one of these triangles?
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