A regular hexagon with sides of 3" is inscribed in a circle. What is the area of a segment formed by a side of the hexagon and the circle?

written in exact form

To find the area of the segment formed by a side of the hexagon and the circle, we need to know the central angle of the sector formed by the side of the hexagon and the center of the circle.

The central angle can be calculated using the formula:

Central angle = 360° / Number of sides of the regular polygon

As the hexagon has 6 sides, the central angle is:

Central angle = 360° / 6 = 60°

Next, we need to find the area of the sector formed by the central angle. The formula to calculate the area of a sector is:

Area of sector = (central angle / 360°) x πr²

Since the side of the hexagon is inscribed in the circle, the radius of the circle is equal to the distance from the center of the circle to one of its vertices. To find this distance, we can draw two radii from the center of the circle to two adjacent vertices of the hexagon, forming an equilateral triangle. The side length of the equilateral triangle is equal to the radius of the circle.

So, the radius of the circle is 3".

Now we can calculate the area of the sector using the formula:

Area of sector = (60° / 360°) x π(3")²

To simplify,

Area of sector = (1/6) x π(3")²

Area of sector = (1/6) x 9π

Area of sector = (3/2)π

Therefore, the area of the segment formed by a side of the hexagon and the circle is (3/2)π square units.