Area and Perimeter of Regular Polygons worksheet - Free Printable

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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## ✔ 1. Pentagon (Top Left)
- Side length: 18 units
- Apothem: 15 units

This is a regular pentagon, so:
- $ n = 5 $
- $ s = 18 $
- $ a = 15 $

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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## ✔ 2. Octagon (Top Right)
- Side length: 10 m
- Apothem: 12.1 m

This is a regular octagon, so:
- $ n = 8 $
- $ s = 10 $
- $ a = 12.1 $

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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## ✔ 3. Octagon (Middle Left)
- Side length: 5 cm
- Apothem: $ a = 6 $ cm

Again, regular octagon:
- $ n = 8 $
- $ s = 5 $
- $ a = 6 $

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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## ✔ 4. Hexagon (Middle Right)
- Side length: 5 cm
- Apothem: $ a = 4.3 $ cm

Regular hexagon:
- $ n = 6 $
- $ s = 5 $
- $ a = 4.3 $

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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## ✔ 5. Pentagon (Bottom Left)
- Side length: 10 cm
- Apothem: $ a = 6.9 $ cm

Regular pentagon:
- $ n = 5 $
- $ s = 10 $
- $ a = 6.9 $

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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## ✔ 6. Pentagon (Bottom Right)
- Side length: 2 cm
- Apothem: $ a = 1.4 $ cm

Regular pentagon:
- $ n = 5 $
- $ s = 2 $
- $ a = 1.4 $

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 (1) | 90 units | 675 sq units |
| Octagon (2) | 80 m | 484 m² |
| Octagon (3) | 40 cm | 120 cm² |
| Hexagon (4) | 30 cm | 64.5 cm² |
| Pentagon (5) | 50 cm | 172.5 cm² |
| Pentagon (6) | 10 cm | 7 cm² |

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✔ All calculations are based on standard formulas for regular polygons using side length and apothem.

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