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Math Help - 9-sided polygon circumscribed around a circle

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    9-sided polygon circumscribed around a circle

    if a regular 9-sided polygon was circumscribed around a circle, what is the difference between the areas of the polygons
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    Area problem

    Hello tomatoes
    Quote Originally Posted by tomatoes View Post
    if a regular 9-sided polygon was circumscribed around a circle, what is the difference between the areas of the polygons
    You don't say what the radius of the circle is (so I'll call it r), and I assume that you want the difference between the area of the polygon and the area of the circle.

    So, imagine the polygon made up of nine isosceles triangles, with a common vertex at the centre of the circle. Then cut one of these triangles in half by a line from the centre to the point where the side of the polygon touches the circle. This line is r units long, and the angle at the centre of the circle in this triangle is \frac{360}{18}= 20^o. From this, you should be able to see that the lines joining the centre of the circle to the vertices of the polygon have length \frac{r}{\cos20^o}.

    Using the formula for the area of a triangle \tfrac12bc\sin A, this gives the area of one of the nine original triangles as \frac12\Big(\frac{r}{\cos20^o}\Big)^2\sin 40^o.

    Multiply this by 9, and subtract \pi r^2, and you're done. OK?

    Grandad
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