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 .

The area of a regular polygon is equal to one-half the product of its apothem - the perpendicular distance from the center to a side - and its perimeter. 

The perimeter of the octagon is 

From the diagram below, the apothem of the octagon is . 

 is one half of the sidelength, or 5.  can be seen to be the length of a leg of a  triangle with hypotenuse 10, or

This makes the apothem . 

The area is therefore 

A regular hexagon can be seen as a composite of six equilateral triangles, each of whose sidelength is the sidelength of the hexagon:

Each of the triangles has area

Substitute  to get 

Multiply this by 6: , the area of the hexagon.

This figure can be seen as a composite of two simple shapes: the rectangle with vertices , and the triangle with vertices .

The rectangle has length  and height , so its area is the product of these dimensions, or .

The triangle has as its base the length of the horizontal segment connecting  and , which is ; its height is the vertical distance from the other vertex to this segment, which is . The area of this triangle is half the product of the base and the height, which is .

Add the areas of the rectangle and the triangle to get the total area:

This figure can be seen as a smaller rectangle cut out of a larger one; refer to the diagram below.

We can fill in the missing sidelengths using the fact that a rectangle has congruent opposite sides. Once this is done, we can multiply length times height of both rectangles to get the area of each, and subtract areas:

 square feet