geometry problem?

Question

if a regular hexagon is inscribed in a circle with a radius of 25". find the length of each side of the hexagon.
steps and work please

Answer 1

The length of each side is 25" !

That's fundamentally because a hexagon has 6 sides. The argument proceeds via the ESSENTIAL first step of PROVING (not just asserting) that a hexagon can be viewed as "being made up from six equilateral triangles." The following analysis does just that, on the way to the final determination.

If you draw a line joining each vertex to the circle's centre, there are 6 of them, equally spaced in angle. So the angles between these adjacent lines, at the circle's centre, are all 60 degrees. All the adjacent radii have the same length (25"), so that each of the six triangles now making up the hexagon are not merely isosceles triangles (two sides having the same length of 25", since they are both radii of the circle), they are also equilateral triangles (because there's a 60 degree angle between those two equal radii in each separate triangle). Therefore all angles in these triangles are equal --- at 60 degrees --- therefore you have six equilateral triangles making up the hexagon. Since they're equilateral, their THIRD sides, making up the sides of the hexagon, are therefore of length 25" also.

QED

Live long and prosper.

Answer 2

The answer is 25". If a hexagon is inscribed in the circle, then notice that the line going from the center of the hexagon to any vertex of the hexagon is also a radius of the circle. If you draw all of the lines liike this from the center to each of the 6 points, you'll find that they split the hexagon into 6 triangles.

Pick one of these triangles. Notice that the angle in between the radii is 360/6 = 60 degrees. This has to be an isoceles triangle since it's sides on either side of the 60 degree angle is r=25". So the remaining angles each have the same measure x, where also 60 + x + x =180. Solve this for x, and you get x = 60. Therefore, each of the little 6 triangles are equilateral. So the edge of the hexagon (which is on every triangle) must be 25" too.

Answer 3

A regular hexagon can be cut into 6 equilateral triangles. Therefore, the length of each side equals to the radius of the circle.

Answer 4

The center angle of an hexagon is 60 degrees.One side with the radii corresponding to its vertex form an euilateral triangle .So each side equals the radium =25"