A regular 12-sided polygon is inscribed in a circle of a radius r cm. Find the area (in sq.cm) of the region between the circle and the polygon.

To solve this problem, we first need to find the area of the circle and the area of the regular 12-sided polygon (also known as a dodecagon), then subtract the latter from the former.

Step 1: Find the area of the circle The formula for the area of a circle is πr². So, the area of the circle is πr² square cm.

Step 2: Find the area of the dodecagon A regular dodecagon can be divided into 12 congruent isosceles triangles. The central angle for each triangle is 360°/12 = 30°. Each triangle has a base of length r (the radius of the circle) and a height which can be found using trigonometry (specifically, the sine of the half central angle). The area of each triangle is 1/2 * base * height, and the total area of the dodecagon is 12 times the area of one triangle.

The height of each triangle is rsin(15°). So, the area of each triangle is 1/2 * r * rsin(15°) = r²sin(15°)/2 square cm. The total area of the dodecagon is 12r²sin(15°)/2 = 6r²*sin(15°) square cm.

Step 3: Find the area of the region between the circle and the dodecagon Subtract the area of the dodecagon from the area of the circle to get the area of the region between them. This is πr² - 6r²sin(15°) square cm.