Lesson: Challenging Problem Solving - 32t01

Multiple Figures: Practice

This question asks us to find the ratio of the area of the circle to the area of the polygon. To do so, we will need to determine the radius of the circle. We will also need to figure out a way to determine the area of the polygon; since polygons are composed of more familiar figures such as triangles and quadrilaterals, a good approach will be to break up the polygon into these more manageable shapes.

Now, what are we given? We are told that the circle is inscribed in the polygon. Therefore, we know that 1) the sides of the polygon must be equal , 2) the line segments that join opposite angles of the polygon are equal in length, and they also intersect at the circle's center, and 3) the circle's radii form right angles with the polygon sides at the point of tangency.

Therefore, we have a polygon composed of 6 equilateral triangles; because the outer leg of each triangle is 6, the angles that converge at center of circle must also be equal . The two other angles in each triangle must also be equal since their opposite sides are equal .

Our 6 equilateral triangles can be further broken down into 12 30°/60°/90° triangles formed by the circle radii. Now, since we have the length of one side of a 30°/60°/90° triangle we can determine the radius of the circle by using ratio lengths of a 30:60:90 triangle. Once we have radius we can use it to find area of each triangle, and then sum them to get the area of the polygon.

Circle Radius & Area:
A 30°/60°/90° triangle has side lengths in the ratio of . If the shortest side of this type of triangle is 3, then the second longest side must be . Therefore, the circle's radius measures , and the circle's area is

.

Polygon Area:
If the circle's radius is , then the area of each 30°/60°/90° triangle is . The polygon is composed of 12 30/60/90 triangles, so the polygon area is .

The ratio of the area of the circle to the area of the polygon is , or .

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