A polygon with eight sides is called an octagon.

In naming a polygon, the vertices must be written in the order in which they are positioned, going either clockwise or counterclockwise. Of the four choices, only Polygon $\displaystyle DFEC$ violates this convention, since $\displaystyle D$ and $\displaystyle F$ are not adjacent vertices (nor are $\displaystyle E$ and $\displaystyle C$).

The figure can be viewed as the composite of rectangles. As such, we can take advantage of the fact that opposite sides of a rectangle have the same length, as follows:

Now that the missing sidelengths are known, we can add the sidelengths to find the perimeter:

$\displaystyle P = 40 + 72 + 10 + 22+ 30 + 50 = 224$

A diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon. Of the four choices, only $\displaystyle \overline{KM}$ fits this description.

A polygon with eight sides is called an octagon.

A polygon with six sides is called a hexagon.

The figure described is below. 

Since the hexagon is regular, its sides are congruent, and its angles each have measure $\displaystyle 180 ^{\circ } \times \frac{6-2}{6} = 180 ^{\circ } \times \frac{4}{6} = 120^{\circ }$.

Also, each of the triangles are isosceles, and their acute angles measure $\displaystyle \frac{1}{2} (180^{\circ }- 120^{\circ }) = 30^{\circ }$ each. This means that each of the four angles of Quadrilateral $\displaystyle EXGO$ measures $\displaystyle 120 ^{\circ }- 30^{\circ }= 90^{\circ }$, so Quadrilateral $\displaystyle EXGO$ is a rectangle. However, not all sides are congruent, so it is not a rhombus. Also, since it is a rectangle, it cannot be a trapezoid.

The correct response is I only.

A regular octagon has eight sides of equal length, so multiply the length of one side by eight:

$\displaystyle 6 \frac{1}{4} \times 8 = \frac{25}{4} \times \frac{8}{1} = \frac{200}{4} = 50$ feet.

Divide by three to get the equivalent in yards:

$\displaystyle 50 \div 3 = 16 \frac{2}{3}$ yards.

Write the formula for the area of a semicircle.

$\displaystyle A = \frac{1}{2}\pi r^2$

Substitute the area.

$\displaystyle 16\pi = \frac{1}{2}\pi r^2$

Multiply by 2, and divide by pi on both sides.

$\displaystyle \frac{16\pi \cdot (2)}{\pi}=\frac{ \frac{1}{2}\cdot (2)\pi r^2}{\pi}$

The equation becomes:

$\displaystyle r^2 =32$

Square root both sides and factor the right side.

$\displaystyle \sqrt{r^2} =\sqrt{32}$

$\displaystyle r=4\sqrt{2}$

The diameter is double the radius.

$\displaystyle D=8\sqrt{2}$

The circumference is half the circumference of a full circle.

$\displaystyle C_{semi}=\frac{1}{2}\pi D = 4\pi \sqrt{2}$

The perimeter is the sum of the diameter and the half circumference.

The answer is:  $\displaystyle 8\sqrt2+ 4\pi \sqrt2$

A hexagon has 6 equal sides. The formula to find perimeter of a hexagon is:

$\displaystyle P = 6a$

where a is the length of any side. Now, to find the length of one side, we will solve for a. 

We know the perimeter of the hexagon is 90in. So, we will substitute and solve for a. We get

$\displaystyle 90\text{in} = 6a$

$\displaystyle \frac{90\text{in}}{6} = \frac{6a}{6}$

$\displaystyle 15\text{in} = a$

$\displaystyle a = 15\text{in}$

Therefore, the length of one side of the hexagon is 15in.