Find the area of each regular polygon using the given dimensions.

To find the area of a regular polygon, we generally use this formula:
$$ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} $$

* Perimeter ($P$) is the total distance around the outside (number of sides $\times$ side length).
* Apothem ($a$) is the straight line from the center to the middle of a side (it makes a right angle with the side).
* If the apothem isn't given, but the side length is, we can calculate the apothem using trigonometry: $a = \frac{\text{side}}{2 \times \tan(180^\circ / n)}$, where $n$ is the number of sides.

Let's solve each problem step-by-step.

Problem 2) Regular Pentagon

* Shape: Pentagon ($5$ sides).
* Given: Side length ($s$) = $16$.
* Goal: Find Area.

Step 1: Find the Perimeter.
$$ P = 5 \times 16 = 80 $$

Step 2: Find the Apothem.
Since the apothem isn't drawn, we calculate it.
The angle for one slice of the pie is $360^\circ / 5 = 72^\circ$.
We split that slice in half to make a right triangle, so the angle at the center is $36^\circ$.
The base of this small triangle is half the side length: $16 / 2 = 8$.

Using tangent:
$$ \tan(36^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{8}{a} $$
$$ a = \frac{8}{\tan(36^\circ)} $$
$$ a \approx \frac{8}{0.7265} \approx 11.011 $$

Step 3: Calculate Area.
$$ \text{Area} = \frac{1}{2} \times 80 \times 11.011 $$
$$ \text{Area} = 40 \times 11.011 $$
$$ \text{Area} \approx 440.44 $$

Problem 3) Regular Nonagon

* Shape: Nonagon ($9$ sides).
* Given: Side length ($s$) = $6$.
* Goal: Find Area.

Step 1: Find the Perimeter.
$$ P = 9 \times 6 = 54 $$

Step 2: Find the Apothem.
Center angle for one slice: $360^\circ / 9 = 40^\circ$.
Half-angle for the right triangle: $20^\circ$.
Base of the triangle (half-side): $6 / 2 = 3$.

Using tangent:
$$ \tan(20^\circ) = \frac{3}{a} $$
$$ a = \frac{3}{\tan(20^\circ)} $$
$$ a \approx \frac{3}{0.3640} \approx 8.242 $$

Step 3: Calculate Area.
$$ \text{Area} = \frac{1}{2} \times 54 \times 8.242 $$
$$ \text{Area} = 27 \times 8.242 $$
$$ \text{Area} \approx 222.53 $$

Problem 4) Regular Octagon

* Shape: Octagon ($8$ sides).
* Given: Apothem ($a$) = $4$. Side length is NOT given.
* Goal: Find Area.

Step 1: Find the Side Length.
We need the perimeter, so we must find the side length first using the apothem.
Center angle for one slice: $360^\circ / 8 = 45^\circ$.
Half-angle: $22.5^\circ$.

Using tangent on the half-triangle:
$$ \tan(22.5^\circ) = \frac{\text{half-side}}{4} $$
$$ \text{half-side} = 4 \times \tan(22.5^\circ) $$
$$ \text{half-side} \approx 4 \times 0.4142 = 1.6568 $$
Full side length ($s$) = $1.6568 \times 2 = 3.3136$

Step 2: Find the Perimeter.
$$ P = 8 \times 3.3136 = 26.5088 $$

Step 3: Calculate Area.
$$ \text{Area} = \frac{1}{2} \times 26.5088 \times 4 $$
$$ \text{Area} = 26.5088 \times 2 $$
$$ \text{Area} \approx 53.02 $$

Problem 5) Regular Decagon

* Shape: Decagon ($10$ sides).
* Given: Apothem ($a$) = $8$. Side length is NOT given.
* Goal: Find Area.

Step 1: Find the Side Length.
Center angle for one slice: $360^\circ / 10 = 36^\circ$.
Half-angle: $18^\circ$.

Using tangent:
$$ \tan(18^\circ) = \frac{\text{half-side}}{8} $$
$$ \text{half-side} = 8 \times \tan(18^\circ) $$
$$ \text{half-side} \approx 8 \times 0.3249 = 2.5992 $$
Full side length ($s$) = $2.5992 \times 2 = 5.1984$

Step 2: Find the Perimeter.
$$ P = 10 \times 5.1984 = 51.984 $$

Step 3: Calculate Area.
$$ \text{Area} = \frac{1}{2} \times 51.984 \times 8 $$
$$ \text{Area} = 51.984 \times 4 $$
$$ \text{Area} \approx 207.94 $$

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Final Answer:
2) 440.44 square units
3) 222.53 square units
4) 53.02 square units
5) 207.94 square units