Find the area and perimeter of regular polygons with given dimensions.

Let's solve each of the regular polygons step by step. We'll calculate both perimeter and area for each shape using the given information.

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🔷 General Formulas for Regular Polygons

For a regular polygon with:
- $ n $ sides
- side length $ s $
- apothem $ a $ (distance from center to midpoint of a side)

Then:
- Perimeter = $ P = n \times s $
- Area = $ A = \frac{1}{2} \times P \times a = \frac{1}{2} \times n \times s \times a $

We’ll use these formulas throughout.

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## ✔ Problem 1: Regular Pentagon (Top Left)

Given:
- Side length $ s = 18 $ units
- Apothem $ a = 15 $ units
- Number of sides $ n = 5 $

🔹 Perimeter:

$$
P = 5 \times 18 = 90 \text{ units}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 90 \times 15 = 45 \times 15 = 675 \text{ square units}
$$

✔ Answer:
- Perimeter: 90 units
- Area: 675 square units

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## ✔ Problem 2: Regular Octagon (Top Right)

Given:
- Side length $ s = 10 $ m
- Apothem $ a = 12.1 $ m
- Number of sides $ n = 8 $

🔹 Perimeter:

$$
P = 8 \times 10 = 80 \text{ m}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 80 \times 12.1 = 40 \times 12.1 = 484 \text{ m}^2
$$

✔ Answer:
- Perimeter: 80 m
- Area: 484 m²

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## ✔ Problem 3: Regular Octagon (Middle Left)

Given:
- Side length $ s = 5 $ cm
- Apothem $ a = 6 $ cm
- $ n = 8 $

🔹 Perimeter:

$$
P = 8 \times 5 = 40 \text{ cm}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 40 \times 6 = 20 \times 6 = 120 \text{ cm}^2
$$

✔ Answer:
- Perimeter: 40 cm
- Area: 120 cm²

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## ✔ Problem 4: Regular Hexagon (Middle Right)

Given:
- Side length $ s = 5 $ cm
- Apothem $ a = 4.3 $ cm
- $ n = 6 $

🔹 Perimeter:

$$
P = 6 \times 5 = 30 \text{ cm}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 30 \times 4.3 = 15 \times 4.3 = 64.5 \text{ cm}^2
$$

✔ Answer:
- Perimeter: 30 cm
- Area: 64.5 cm²

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## ✔ Problem 5: Regular Pentagon (Bottom Left)

Given:
- Side length $ s = 10 $ cm
- Apothem $ a = 6.9 $ cm
- $ n = 5 $

🔹 Perimeter:

$$
P = 5 \times 10 = 50 \text{ cm}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 50 \times 6.9 = 25 \times 6.9 = 172.5 \text{ cm}^2
$$

✔ Answer:
- Perimeter: 50 cm
- Area: 172.5 cm²

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## ✔ Problem 6: Regular Pentagon (Bottom Right)

Given:
- Side length $ s = 2 $ cm
- Apothem $ a = 1.4 $ cm
- $ n = 5 $

🔹 Perimeter:

$$
P = 5 \times 2 = 10 \text{ cm}
$$

🔹 Area:

$$
A = \frac{1}{2} \times 10 \times 1.4 = 5 \times 1.4 = 7 \text{ cm}^2
$$

✔ Answer:
- Perimeter: 10 cm
- Area: 7 cm²

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📌 Final Answers Summary:

| Shape | Perimeter | Area |
|------|-----------|------|
| Pentagon (top left) | 90 units | 675 sq units |
| Octagon (top right) | 80 m | 484 m² |
| Octagon (middle left) | 40 cm | 120 cm² |
| Hexagon (middle right) | 30 cm | 64.5 cm² |
| Pentagon (bottom left) | 50 cm | 172.5 cm² |
| Pentagon (bottom right) | 10 cm | 7 cm² |

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Let me know if you'd like a visual explanation or help understanding the apothem concept!