The measure of each interior angle of a regular polygon with $\displaystyle N$ sides is $\displaystyle \frac{180 (N-2)}{N}$. We can substitute $\displaystyle N = 23$ to obtain the angle measure:

$\displaystyle \frac{180 (N-2)}{N} = \frac{180 (23-2)}{23} = \frac{180 \cdot 21}{23}\approx 164.3^{\circ }$

The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure $\displaystyle 180-162 = 18^{\circ }$. Since 360 divided by the number of sides of a regular polygon is equal to the measure of one of its exterior angles, we are seeking $\displaystyle N$ such that 

$\displaystyle \frac{360}{N} = 18$

Solve for $\displaystyle N$:

$\displaystyle \frac{360}{N}\cdot N= 18\cdot N$

$\displaystyle 18N = 360$

$\displaystyle 18N \div 18= 360\div 18$

$\displaystyle N = 20$

The polygon has 20 sides.

To find the side length from the area of an octagon and the apothem we must use the area of a polygon which is

First plug in our numbers for area and the apothem to get 

$\displaystyle 160=\frac{1}{2}(8)(perimeter)$

Then multiply to get $\displaystyle 160=4(perimeter)$

Then divide both sides by 4 to get the perimeter of the figure. $\displaystyle \frac{160}{4}=40$

When we have the perimeter of a regular polygon, to find the side length we must divide by the number of sides of the polygon, in this case 8. $\displaystyle \frac{40}{8}=5$

After dividing we find the side length is  $\displaystyle 5.$

Proporations can be used to solve for the central angle. Let $\displaystyle x$ equal the angle of the sector.

$\displaystyle \frac{20}{100}=\frac{x}{360}$

Cross mulitply:

$\displaystyle 7200=100x$

Solve for $\displaystyle x$:

$\displaystyle \frac{7200}{100}=\frac{100x}{100}$

$\displaystyle 72=x$

When two chords of a circle intersect, the measure of the angle they form is half the sum of the measures of the arcs they intercept. Therefore, 

$\displaystyle m \angle AXB = \frac{m\; \widehat{AB} + m\; \widehat{CD} }{2}$

Since $\displaystyle \angle AXB$ and $\displaystyle \angle AXC$ form a linear pair, $\displaystyle m \angle AXB + m \angle AXC = 180$, and $\displaystyle m \angle AXB = 180 - m \angle AXC = 180 - 135 = 45^{\circ }$.

Substitute  $\displaystyle m\; \widehat{AB} = 24$ and $\displaystyle m \angle AXB = 45$ into the first equation:

$\displaystyle 45 = \frac{24 + m\; \widehat{CD} }{2}$

$\displaystyle 90 =24 + m\; \widehat{CD}$

$\displaystyle 66 = m\; \widehat{CD}$

First, let's calculate how many slices there are per pizza. This is done by dividing 360° by the respective slice degrees:

Pizza 1: 360/30 = 12 slices

Pizza 2: 360/45 = 8 slices

Now, the total amount made per pizza is calculated by multiplying the number of slices by the respective cost per slice:

Pizza 1: 12 * 1.95 = $23.40

Pizza 2: 8 * 2.25 = $18.00

The difference between Pizza 2 and Pizza 1 is thus represented by: 18 – 23.40 = –$5.40

In order to find the percentage of a sector from an angle, you need to know that a full circle is $\displaystyle 360^{\circ}$.

Therefore, we can find the percentage by dividing the angle of the sector by $\displaystyle 360^{\circ}$ and then multiplying by 100:

$\displaystyle Percentage = \frac{\measuredangle}{360^{\circ}}\cdot 100$

$\displaystyle Percentage = \frac{60^{\circ}}{360^{\circ}}\cdot 100=\frac{1}{6}\cdot 100=.1\bar{6}\cdot 100=16.7\%$

Proportions can be used to determine the percentage. Let $\displaystyle x$ equal the percentage comprised of the sector.

$\displaystyle \frac{36}{360}=\frac{x}{100}$

Cross multiply

$\displaystyle 3600=360x$

Solve for x.

$\displaystyle \frac{3600}{360}=\frac{360x}{360}$

$\displaystyle 10=x$

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s². Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16).

Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 6² = 36π in². If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in².

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

$\displaystyle AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20$

Since AB is a radius of B, we can find the area of circle B via:

$\displaystyle Area=\pi R^2=\pi(20^2)=400\pi$

Angle DBE is a right angle, and therefore $\displaystyle \frac{1}{4}$ of the circle so it follows:

$\displaystyle Area(DBE)=\frac{400}{4}\pi=100\pi$