Final answer:
Final answer:
The ratio of the area of a circle inscribed in a regular hexagon to that of the hexagon is obtained by dividing the area of the circle (πr^2) by the area of the hexagon (3r^2*sqrt(3)) resulting in the ratio π / (3sqrt(3)).
Step-by-step explanation:
To find the ratio of the area of a circle to that of a regular hexagon in which it is inscribed, we can begin by determining the relationship between the side length of the hexagon and the radius of the circle. In a regular hexagon, a circle inscribed within it will touch all six sides at their midpoints, meaning the radius of the circle (r) is equal to the side length of the hexagon (a).
Using this information, the area of the circle is πr2 and the area of the hexagon can be found by dividing it into six equilateral triangles, each with an area of ½*a2*√3. Since there are six triangles, the total area of the hexagon is 6*½*a2*√3 = 3a2*√3. As r = a, this can be rewritten as 3r2*√3.
Now, to find the ratio of the area of the circle to the area of the hexagon, we divide the area of the circle by the area of the hexagon, which gives us πr2 / (3r2*√3). Simplifying this, we end up with the ratio π / (3√3), which cannot be further simplified and represents the desired ratio of areas.