Find the perimeter and apothem of each polygon based on given area and side length.

Let's solve each problem step by step using the formula for the area of a regular polygon:

$$
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
$$

We can rearrange this formula to solve for perimeter and apothem:

- $ \text{Perimeter} = \frac{2 \times \text{Area}}{\text{Apothem}} $
- $ \text{Apothem} = \frac{2 \times \text{Area}}{\text{Perimeter}} $

But in these problems, we are given:
- The area
- The side length (or implied from shape and label)
- And the number of sides (from the shape)

So we’ll use:
- Perimeter = number of sides × side length
- Then use the area formula to find the apothem.

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Let’s go through each one.

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1) Pentagon

- Side = 7.5 yd
- Area = 337.21 yd²
- Number of sides = 5

Perimeter = $ 5 \times 7.5 = 37.5 $ yd

Now use area formula:

$$
\text{Area} = \frac{1}{2} \times P \times a \Rightarrow 337.21 = \frac{1}{2} \times 37.5 \times a
$$

$$
337.21 = 18.75a \Rightarrow a = \frac{337.21}{18.75} \approx 18.00
$$

✔ Perimeter: 37.50 yd, Apothem: 18.00 yd

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2) Hexagon

- Side = 6 ft
- Area = 93.53 ft²
- Sides = 6

Perimeter = $ 6 \times 6 = 36 $ ft

$$
93.53 = \frac{1}{2} \times 36 \times a = 18a \Rightarrow a = \frac{93.53}{18} \approx 5.196 \approx 5.20
$$

✔ Perimeter: 36.00 ft, Apothem: 5.20 ft

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3) Octagon

- Side = 11 in
- Area = 439.7 in²
- Sides = 8

Perimeter = $ 8 \times 11 = 88 $ in

$$
439.7 = \frac{1}{2} \times 88 \times a = 44a \Rightarrow a = \frac{439.7}{44} \approx 9.993 \approx 9.99
$$

✔ Perimeter: 88.00 in, Apothem: 9.99 in

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4) Triangle (Equilateral)

- Side = 8 ft
- Area = 35.07 ft²
- Sides = 3

Perimeter = $ 3 \times 8 = 24 $ ft

$$
35.07 = \frac{1}{2} \times 24 \times a = 12a \Rightarrow a = \frac{35.07}{12} \approx 2.9225 \approx 2.92
$$

✔ Perimeter: 24.00 ft, Apothem: 2.92 ft

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5) Octagon

- Side = 13 in
- Area = 1,044.73 in²
- Sides = 8

Perimeter = $ 8 \times 13 = 104 $ in

$$
1044.73 = \frac{1}{2} \times 104 \times a = 52a \Rightarrow a = \frac{1044.73}{52} \approx 20.091 \approx 20.09
$$

✔ Perimeter: 104.00 in, Apothem: 20.09 in

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6) Heptagon (7 sides)

- Side = 8 yd
- Area = 309.01 yd²
- Sides = 7

Perimeter = $ 7 \times 8 = 56 $ yd

$$
309.01 = \frac{1}{2} \times 56 \times a = 28a \Rightarrow a = \frac{309.01}{28} \approx 11.036 \approx 11.04
$$

✔ Perimeter: 56.00 yd, Apothem: 11.04 yd

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7) Nonagon (9 sides)

- Side = 3 in
- Area = 192.36 in²
- Sides = 9

Perimeter = $ 9 \times 3 = 27 $ in

$$
192.36 = \frac{1}{2} \times 27 \times a = 13.5a \Rightarrow a = \frac{192.36}{13.5} \approx 14.249 \approx 14.25
$$

✔ Perimeter: 27.00 in, Apothem: 14.25 in

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8) Pentagon

- Side = 20 yd
- Area = 688.15 yd²
- Sides = 5

Perimeter = $ 5 \times 20 = 100 $ yd

$$
688.15 = \frac{1}{2} \times 100 \times a = 50a \Rightarrow a = \frac{688.15}{50} = 13.763 \approx 13.76
$$

✔ Perimeter: 100.00 yd, Apothem: 13.76 yd

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9) Square

- Side = 12 ft
- Area = 144 ft²
- Sides = 4

Perimeter = $ 4 \times 12 = 48 $ ft

$$
144 = \frac{1}{2} \times 48 \times a = 24a \Rightarrow a = \frac{144}{24} = 6.00
$$

✔ Perimeter: 48.00 ft, Apothem: 6.00 ft

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✔ Final Answers (Rounded to Two Decimal Places):

| Problem | Perimeter | Apothem |
|--------|-----------|---------|
| 1) | 37.50 yd | 18.00 yd |
| 2) | 36.00 ft | 5.20 ft |
| 3) | 88.00 in | 9.99 in |
| 4) | 24.00 ft | 2.92 ft |
| 5) | 104.00 in | 20.09 in |
| 6) | 56.00 yd | 11.04 yd |
| 7) | 27.00 in | 14.25 in |
| 8) | 100.00 yd | 13.76 yd |
| 9) | 48.00 ft | 6.00 ft |

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🔍 Explanation Summary:

For regular polygons, you can always use:
- Perimeter = n × s (n = number of sides, s = side length)
- Area = (1/2) × Perimeter × Apothem
- So: Apothem = (2 × Area) / Perimeter

This method works because the apothem is the distance from the center to the midpoint of a side, and it acts as the height of each of the congruent triangles that make up the polygon.

Let me know if you'd like a visual explanation or formulas!