Area of a Regular Polygon worksheet with six geometric shapes and measurement data.

To find the area of each regular polygon, we use the formula:

> Area = (Perimeter × Apothem) ÷ 2

But since we’re given side lengths and sometimes apothems (the perpendicular distance from center to a side), we need to figure out which formula applies.

Actually, for regular polygons, if you know the side length and the number of sides, you can use:

> Area = (n × s²) / (4 × tan(π/n))

Where:
- n = number of sides
- s = side length
- tan is tangent function (use calculator in degree mode)

BUT — looking at the diagrams, some give the apothem (like #1, #3, #5, #6), and others give the side length (#2, #4).

Wait — let’s check each one carefully.

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Problem 1: Square with side = 4½ in = 4.5 in

For a square, area = side × side
→ 4.5 × 4.5 = 20.25 → already 2 decimal places.

✔ Answer: 20.25 in²

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Problem 2: Equilateral triangle with side = 12 cm

Formula for area of equilateral triangle:
> Area = (√3 / 4) × side²

So:
(1.73205 / 4) × 144 ≈ 0.4330125 × 144 ≈ 62.3538 → round to 2 decimals → 62.35 cm²

✔ Answer: 62.35 cm²

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Problem 3: Regular hexagon with apothem = 3 ft

For a regular hexagon, if you know the apothem (a), then:

> Area = (Perimeter × Apothem) / 2

But we don’t have perimeter yet.

In a regular hexagon, the apothem relates to the side length by:

> a = (s × √3) / 2 → so s = (2a)/√3

Given a = 3 ft:

s = (2×3)/√3 = 6/1.73205 ≈ 3.4641 ft

Perimeter = 6 × s ≈ 6 × 3.4641 ≈ 20.7846 ft

Now area = (P × a)/2 = (20.7846 × 3)/2 ≈ 62.3538 / 2? Wait no:

Wait: (20.7846 × 3) = 62.3538 → divided by 2 = 31.1769 → that can’t be right because earlier I thought it was ~62.

Wait — I think I messed up.

Actually, for a regular hexagon, there's a simpler formula using apothem:

> Area = (3√3 × s²) / 2 — but we don’t have s.

Alternatively, since we have apothem, and for hexagon, the area can also be calculated as:

> Area = (1/2) × Perimeter × Apothem

We found s ≈ 3.4641 ft → P = 6 × 3.4641 ≈ 20.7846 ft

Then Area = (20.7846 × 3) / 2 = 62.3538 / 2 = 31.1769 → that seems too small.

Wait — let me double-check.

Actually, in a regular hexagon, the apothem is the height of one of the 6 equilateral triangles that make up the hexagon.

Each triangle has base = s, height = apothem = 3 ft.

Area of one triangle = (base × height)/2 = (s × 3)/2

Total area = 6 × (s × 3)/2 = 9s

But we still need s.

From geometry: in a regular hexagon, apothem a = (s√3)/2 → so s = 2a/√3 = 6/√3 = 2√3 ≈ 3.4641 ft

Then area = 9s = 9 × 3.4641 ≈ 31.1769 → same as before.

But that feels off. Let me calculate numerically:

s = 2 * 3 / sqrt(3) = 6 / 1.7320508 ≈ 3.4641016

Perimeter = 6 * 3.4641016 = 20.7846096

Area = (P * a)/2 = (20.7846096 * 3)/2 = 62.3538288 / 2 = 31.1769144 → rounds to 31.18 ft²

But wait — I recall that for a regular hexagon with side s, area is (3√3/2)s²

Let me try that:

s = 2√3 ≈ 3.4641

s² = (2√3)^2 = 4 * 3 = 12

Area = (3√3 / 2) * 12 = (3*1.73205/2)*12 ≈ (5.19615/2)*12 = 2.598075 * 12 = 31.1769 → same thing.

So yes, 31.18 ft²

But let me confirm with another approach.

A regular hexagon can be divided into 6 equilateral triangles.

If apothem is 3, then each triangle has height 3.

In an equilateral triangle, height h = (√3/2) * side → so side = 2h/√3 = 6/√3 = 2√3, same as above.

Area of one triangle = (√3/4) * side² = (1.73205/4) * 12 ≈ 0.4330125 * 12 = 5.19615

Times 6 = 31.1769 → yes.

So final answer for #3: 31.18 ft²

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Problem 4: Regular pentagon with side = 7 in

Formula: Area = (n × s²) / (4 × tan(π/n))

n=5, s=7

First, π/5 radians = 36 degrees

tan(36°) ≈ 0.7265425

So denominator: 4 × 0.7265425 ≈ 2.90617

Numerator: 5 × 49 = 245

Area = 245 / 2.90617 ≈ 84.303 → round to 2 decimals → 84.30 in²

Let me calculate more precisely:

tan(36°) = tan(36) = approximately 0.726542528

4 * tan(36) = 2.906170112

245 / 2.906170112 ≈ 84.30308 → yes, 84.30 in²

✔ Answer: 84.30 in²

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Problem 5: Regular octagon with apothem = 6 cm

Regular octagon, n=8, apothem a=6 cm

Formula: Area = (1/2) × Perimeter × Apothem

Need perimeter.

In a regular octagon, apothem a = (s/2) × cot(π/8)

Or: s = 2a × tan(π/8)

π/8 = 22.5 degrees

tan(22.5°) = √2 - 1 ≈ 0.414213562

So s = 2 × 6 × 0.414213562 ≈ 12 × 0.414213562 ≈ 4.970562744 cm

Perimeter = 8 × s ≈ 8 × 4.970562744 ≈ 39.76450195 cm

Area = (P × a)/2 = (39.76450195 × 6)/2 = 238.5870117 / 2 = 119.29350585 → round to 119.29 cm²

Alternatively, use direct formula:

Area = 2 × (1 + √2) × s² — but we don't have s.

Or using apothem:

General formula for regular polygon: Area = (n × a² × tan(π/n))

Wait, let's derive:

Since a = (s/2) / tan(π/n) → s = 2a tan(π/n)

Then Area = (n × s²) / (4 tan(π/n)) = [n × (4a² tan²(π/n))] / [4 tan(π/n)] = n × a² × tan(π/n)

Yes! So Area = n × a² × tan(π/n)

For octagon, n=8, a=6, π/n = 22.5°

tan(22.5°) ≈ 0.414213562

Area = 8 × 36 × 0.414213562 = 288 × 0.414213562 ≈ 119.293505856 → same as before → 119.29 cm²

✔ Answer: 119.29 cm²

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Problem 6: Regular decagon with apothem = 11 cm

n=10, a=11 cm

Use formula: Area = n × a² × tan(π/n)

π/n = 18 degrees

tan(18°) ≈ 0.324919696

Area = 10 × (121) × 0.324919696 = 1210 × 0.324919696 ≈ ?

Calculate: 1210 × 0.324919696

First, 1200 × 0.324919696 = 389.9036352

10 × 0.324919696 = 3.24919696

Total = 389.9036352 + 3.24919696 = 393.15283216 → round to 393.15 cm²

Double-check:

1210 × 0.324919696

Do 1210 × 0.3 = 363

1210 × 0.024 = 29.04

1210 × 0.000919696 ≈ 1210 × 0.0009 = 1.089, plus 1210×0.000019696≈0.0238 → total approx 1.1128

So 363 + 29.04 = 392.04 + 1.1128 ≈ 393.1528 → yes.

✔ Answer: 393.15 cm²

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Final Answers:

1) 20.25 in²
2) 62.35 cm²
3) 31.18 ft²
4) 84.30 in²
5) 119.29 cm²
6) 393.15 cm²

All rounded to 2 decimal places as required.

Final Answer:
1) 20.25 in²
2) 62.35 cm²
3) 31.18 ft²
4) 84.30 in²
5) 119.29 cm²
6) 393.15 cm²