A regular dodecagon is a polygon with twelve sides of equal length, so if one side has a length of , then the perimeter will be equal to twelve times the length of that one side. This gives us:

 is a diameter of the regular hexagon. Examine the diagram below, which shows the hexagon with all three diameters:

Each interior angle of a hexagon measures , so, by symmetry, each base angle of the triangle formed is ; also, each central angle measures one sixth of , or . Each triangle is equilateral, so if , it follows that , and .

Construct two other diagonals as shown.

Each of the interior angles of a regular octagon have measure , so it can be shown that  is a 45-45-90 triangle. Its hypotenuse is , whose length is 8, so, by the 45-45-90 Triangle Theorem, the length of  is 8 divided by :

Likewise, .

Since Quadrilateral  is a rectangle, .

By dividing the figure into rectangles and taking advantage of the fact that opposite sides of rectangles are congruent, we have the following sidelengths:

 is the hypotenuse of a triangle with legs of lengths 8 and 16, so its length can be calculated using the Pythagorean Theorem:

The question can now be answered by noting that  and . 

,

so  falls between 17 and 18.

A regular pentagon has five diagonals of equal length, each formed by a line going from one vertex of the pentagon to another. We can see that for one of these diagonals, an isosceles triangle is formed where the two equal side lengths between the vertices joined by the diagonal are the other two sides. If we draw a line bisecting the angle between those two sides perpendicular with the diagonal that forms the other side of the triangle, we will have two congruent right triangles whose hypotenuse is the side length, , and whose adjacent angle is half the measure of one interior angle of a pentagon. Using these two values, we can solve for the length of the opposite side, which is half of the diagonal, so we can them multiply the result by  to calculate the full length of the diagonal. We start by determining the sum of the interior angles of a pentagon using the following formula, where  is the number of sides of the polygon:

So to get the measure of each of the five angles in a pentagon, we divide the result by :

So each interior angle of a regular pentagon has a measure of . As explained earlier, we can find the length of half the diagonal by bisecting this angle to form two right triangles. If the hypotenuse is  and the adjacent angle is half of an interior angle, or , then the length of the opposite side will be the hypotenuse times the sine of that angle. This only gives half of the diagonal, however, as there are two of these congruent right triangles, so we multiply the result by  and we get the full length of the diagonal of a pentagon as follows:

Extend sides  and  as shown to divide the polygon into three rectangles:

Taking advantage of the fact that opposite sides of a rectangle are congruent, we can find  and :

 is right, so by the Pythagorean Theorem,

The measures of the interior angles of a convex pentagon total

,

so

The pentagon referenced is the one below. Note that the diagonal , along with congruent sides  and , form an isosceles triangle .

Now construct the altitude from  to :

 bisects  and  to form two 30-60-90 triangles. Therefore, ,

and .

This regular hexagon can be seen as being made up of six equilateral triangles, each formed by a side and two radii; each has sidelength  centimeters. The area of one triangle is

There are six such triangles, so multiply this by 6:

A quadrilateral with the maximum area, given a specific perimeter, is a square. Since  and a square has four equal sides, the max area is

Let denote the length of one side of a square.  This is also the top of the trapezoid. Let denote the bottom of the trapezoid. Finally, let be the height of the trapezoid.  The area of the trapezoid is then while the area of the square is .

We then have the total area as 100, so:

Now we know that the red line has length 15.  is the region of this line that is in the trapezoid. What we notice, however, is that the remainder is precisely the length of one side of a square.  So or

Rewriting the previous equation:

This is now an equation of 2 variables and we can easily cross out answers by plugging in possible values.  What we find is that for , respectively.  For we get , which is too small ( must be greater than ). For we get .