The perimeter is the sum of the measures of the sidelengths.

Knowing that the hexagon is regular only tells you the six sides are congruent; without the measure of any side, this does not help you. 

Knowing only that each of the six sides measures 10 cm is by itself enough to calculate the perimeter to be

$\displaystyle 6\times 10=60\ cm$.

The answer is that Statement 1 is sufficient, but not Statement 2.

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 add the lengths of the sides to get the perimeter:

$\displaystyle P = 10 + 3 + 15 + 7 + 25 + 10 = 70$ feet.

The perimeter $\displaystyle P$ of any figure is the sum of the lengths of its sides. Since we have a rectangle with a length of $\displaystyle 20cm$ and a width of $\displaystyle 5cm$, we know that there will be two sides of length $\displaystyle 20cm$ and two sides of width $\displaystyle 5cm$. Therefore:

$\displaystyle P=20cm+20cm+5cm+5cm$

$\displaystyle P=40cm+10cm$

$\displaystyle P=50cm$

In order to find the perimeter $\displaystyle P$ of the right triangle, we need to know the lengths of each of its sides. While we are given two sides - the base $\displaystyle (a)$ and the height $\displaystyle (b)$ - we do not know the hypotenuse $\displaystyle (c)$. There are two ways that we can find $\displaystyle c$, the first of which is the direct application of the Pythagorean Theorem: 

$\displaystyle a^{2}+b^{2}=c^{2}$

$\displaystyle 5^{2}+12^{2}=c^{2}$

$\displaystyle 25+144=c^{2}$

$\displaystyle 13=c$

We could have also noted that $\displaystyle 5:12:13$ is a common Pythagorean Triple and deduced the value of $\displaystyle c$ that way.

Now that we have all three side lengths, we can calculate $\displaystyle P$:

$\displaystyle P=5cm+12cm+13cm$

$\displaystyle P=30cm$

Starting with the knowledge that we are dealing with an octagon, an 8-sided figure, we calculate the perimeter $\displaystyle P$ by adding the lengths of all 8 sides. Since we also know that each side measures $\displaystyle 15cm$, we can use multiplication:

$\displaystyle P=8\times15cm$

$\displaystyle P=120cm$

A pentagon is a 5 sided shape. We are given that two sides are 5 inches each.

Side 1 = 5inches

Side 2 = 5 inches

The next two sides are each 1.5 times bigger than the smallest two sides.

$\displaystyle \small 5*1.5=7.5$

Side 3 =Side 4= 7.5 inches

The last side is twice the size of the smallest side, 

Side 5 =10 inches

Add them all up for our perimeter:

5+5+7.5+7.5+10=35 inches long

A regular dodecagon is a polygon with twelve sides of equal length, so if one side has a length of $\displaystyle 7$, then the perimeter will be equal to twelve times the length of that one side. This gives us:

$\displaystyle P=12(7)=84$

$\displaystyle \overline{AD}$ 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 $\displaystyle 120^{\circ }$, so, by symmetry, each base angle of the triangle formed is $\displaystyle 60^{\circ }$; also, each central angle measures one sixth of $\displaystyle 360^{\circ}$, or $\displaystyle 60^{\circ }$. Each triangle is equilateral, so if $\displaystyle AB = 12$, it follows that $\displaystyle AO = OD = 12$, and $\displaystyle AD = AO+OD = 12 + 12 = 24$.

Construct two other diagonals as shown.

Each of the interior angles of a regular octagon have measure $\displaystyle 135^{\circ }$, so it can be shown that $\displaystyle \bigtriangleup ABX$ is a 45-45-90 triangle. Its hypotenuse is $\displaystyle \overline{AB}$, whose length is 8, so, by the 45-45-90 Triangle Theorem, the length of $\displaystyle \overline{AX}$ is 8 divided by $\displaystyle \sqrt{2}$:

$\displaystyle AX= \frac{8}{\sqrt{2}}= \frac{8\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2}$

Likewise, $\displaystyle YD = 4 \sqrt{2}$.

Since Quadrilateral $\displaystyle BCYX$ is a rectangle, $\displaystyle XY = BC = 8$.

$\displaystyle AD = AX+XY+YD = 4\sqrt{2}+ 8 + 4 \sqrt{2}= 8+8\sqrt{2}$

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

$\displaystyle \overline{AC}$ is the hypotenuse of a triangle with legs of lengths 8 and 16, so its length can be calculated using the Pythagorean Theorem:

$\displaystyle AC = \sqrt{(AB)^{2}+(BC)^{2}} = \sqrt{16^{2}+8^{2}} = \sqrt{256+64} = \sqrt{320}$

The question can now be answered by noting that $\displaystyle 17^{2} = 289$ and $\displaystyle 18^{2}= 324$. 

$\displaystyle 17^{2}< 320 < 18^{2}$,

so $\displaystyle \sqrt{320}$ falls between 17 and 18.