Math worksheet for calculating the area of regular polygons using the apothem and perimeter formula.

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

$$
\text{Area} = \frac{1}{2} \times (\text{apothem}) \times (\text{perimeter})
$$

We'll find the perimeter (sum of all side lengths) and multiply it by the apothem, then divide by 2.

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Problem 1: Regular Pentagon

- Side length = 2 cm
- Apothem = 3 cm
- Number of sides = 5

Perimeter = $5 \times 2 = 10$ cm
Area = $\frac{1}{2} \times 3 \times 10 = 15$ cm²

✔ Answer: 15 cm²

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Problem 2: Regular Hexagon

- Side length = 6 in
- Apothem = 3.5 in
- Number of sides = 6

Perimeter = $6 \times 6 = 36$ in
Area = $\frac{1}{2} \times 3.5 \times 36 = 1.75 \times 36 = 63$ in²

✔ Answer: 63 in²

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Problem 3: Regular Octagon

- Side length = 8 m
- Apothem = 5 m
- Number of sides = 8

Perimeter = $8 \times 8 = 64$ m
Area = $\frac{1}{2} \times 5 \times 64 = 2.5 \times 64 = 160$ m²

✔ Answer: 160 m²

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Problem 4: Regular Octagon

- Side length = 30 ft
- Apothem = 22 ft
- Number of sides = 8

Perimeter = $8 \times 30 = 240$ ft
Area = $\frac{1}{2} \times 22 \times 240 = 11 \times 240 = 2640$ ft²

✔ Answer: 2640 ft²

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Problem 5: Triangle (Equilateral)

- Base = 10 m
- Height (apothem?) → Wait: This is a triangle, not a regular polygon with apothem defined as usual.

But here, the height is given as 3.5 m, and base is 10 m.

For a triangle, area = $\frac{1}{2} \times \text{base} \times \text{height}$

So:
$$
\text{Area} = \frac{1}{2} \times 10 \times 3.5 = 5 \times 3.5 = 17.5 \text{ m}^2
$$

⚠️ Note: This is not a regular polygon with apothem in the same sense, but since the formula is given, and it matches triangle area, we use it.

✔ Answer: 17.5 m²

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Problem 6: Regular Octagon

- Side length = 12 ft
- Apothem = 10 ft
- Number of sides = 8

Perimeter = $8 \times 12 = 96$ ft
Area = $\frac{1}{2} \times 10 \times 96 = 5 \times 96 = 480$ ft²

✔ Answer: 480 ft²

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Problem 7: Regular Pentagon

- Side length = 6 m
- Apothem = 4.5 m
- Number of sides = 5

Perimeter = $5 \times 6 = 30$ m
Area = $\frac{1}{2} \times 4.5 \times 30 = 2.25 \times 30 = 67.5$ m²

✔ Answer: 67.5 m²

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Problem 8: Regular Heptagon

- Side length = 13 mm
- Apothem = 11 mm
- Number of sides = 7

Perimeter = $7 \times 13 = 91$ mm
Area = $\frac{1}{2} \times 11 \times 91 = 5.5 \times 91$

Calculate:
$5.5 \times 91 = (5 \times 91) + (0.5 \times 91) = 455 + 45.5 = 500.5$ mm²

✔ Answer: 500.5 mm²

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✔ Final Answers:

| Problem | Area |
|--------|------|
| 1 | 15 cm² |
| 2 | 63 in² |
| 3 | 160 m² |
| 4 | 2640 ft² |
| 5 | 17.5 m² |
| 6 | 480 ft² |
| 7 | 67.5 m² |
| 8 | 500.5 mm² |

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🔍 Summary:

The key idea is to:
1. Find the perimeter (number of sides × side length)
2. Use the apothem (given as perpendicular distance from center to side)
3. Apply:
$$
\text{Area} = \frac{1}{2} \times \text{apothem} \times \text{perimeter}
$$

This works for all regular polygons, including triangles (as shown in #5), even though they are simple.

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