Area of a Polygon worksheet with diagrams and calculation problems.

Let’s solve each polygon area problem step by step. We’ll use the formula given in the example:

> Area = (1/2) × perimeter × apothem
> or written as:
> A = ½ × P × a

Where:
- P = perimeter (sum of all side lengths)
- a = apothem (distance from center to midpoint of a side, shown as the dashed line perpendicular to a side)

We are told to round answers to two decimal places if needed.

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Problem 1: Square with side = 7", apothem = 3.5"

Wait — for a square, the apothem is half the side length? Let’s check:

Actually, in a regular polygon, the apothem is the distance from center to the middle of a side. For a square with side 7", the apothem should be 3.5" — yes, that matches.

Perimeter of square = 4 × side = 4 × 7 = 28"

Area = ½ × 28 × 3.5 = 14 × 3.5 = 49.00

✔ Answer: 49.00

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Problem 2: Regular hexagon with side = 6 cm, apothem = 5.2 cm

Perimeter = 6 sides × 6 cm = 36 cm

Area = ½ × 36 × 5.2 = 18 × 5.2

Calculate: 18 × 5 = 90, 18 × 0.2 = 3.6 → total = 93.6

✔ Answer: 93.60

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Problem 3: Regular octagon with side = 4 m, apothem = 4.8 m

Perimeter = 8 × 4 = 32 m

Area = ½ × 32 × 4.8 = 16 × 4.8

16 × 4 = 64, 16 × 0.8 = 12.8 → total = 76.8

✔ Answer: 76.80

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Problem 4: Regular heptagon (7-gon) with side = 5 ft, apothem = 5.2 ft

Perimeter = 7 × 5 = 35 ft

Area = ½ × 35 × 5.2 = 17.5 × 5.2

Break it down:
17.5 × 5 = 87.5
17.5 × 0.2 = 3.5
Total = 87.5 + 3.5 = 91.0

✔ Answer: 91.00

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Problem 5: Triangle (equilateral?) with base = 8 yd, height (apothem?) = 6.9 yd

Wait — this is tricky. The diagram shows a triangle with a dashed line from center to base — but triangles don’t have an “apothem” unless they’re equilateral and we’re using the inradius.

But looking at the label: it says “6.9 yd” along the dashed line to the base — and the base is labeled 8 yd. Also, the shape is drawn as a triangle with one side labeled 8 yd, and the dashed line going to the middle of that side — so likely, this is meant to be treated as a regular polygon? But a triangle isn't typically called a "regular polygon" in these worksheets unless specified.

Wait — actually, in some contexts, especially in middle school math, they might treat any polygon with equal sides and angles as regular, and give you the apothem even for triangle.

But here’s the key: the formula A = ½ × P × a works for ANY regular polygon. So if this is an equilateral triangle with side 8 yd, then:

Perimeter = 3 × 8 = 24 yd

Apothem = 6.9 yd (given)

Area = ½ × 24 × 6.9 = 12 × 6.9

12 × 6 = 72, 12 × 0.9 = 10.8 → total = 82.8

✔ Answer: 82.80

*(Note: In reality, for an equilateral triangle with side 8, the actual apothem (inradius) is (√3/6)*side ≈ 2.309, not 6.9 — but since the worksheet gives us 6.9, we go with what’s given. This is probably just a made-up number for practice.)*

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Problem 6: Pentagon with side = 6 in, apothem = 4.1 in

Perimeter = 5 × 6 = 30 in

Area = ½ × 30 × 4.1 = 15 × 4.1

15 × 4 = 60, 15 × 0.1 = 1.5 → total = 61.5

✔ Answer: 61.50

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Problem 7: Hexagon with side = 9 mm, apothem = 7.8 mm

Perimeter = 6 × 9 = 54 mm

Area = ½ × 54 × 7.8 = 27 × 7.8

27 × 7 = 189, 27 × 0.8 = 21.6 → total = 210.6

✔ Answer: 210.60

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Problem 8: Octagon with side = 5 cm, apothem = 6 cm

Perimeter = 8 × 5 = 40 cm

Area = ½ × 40 × 6 = 20 × 6 = 120

✔ Answer: 120.00

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Problem 9: Nonagon (9-gon) with side = 4 in, apothem = 5.5 in

Perimeter = 9 × 4 = 36 in

Area = ½ × 36 × 5.5 = 18 × 5.5

18 × 5 = 90, 18 × 0.5 = 9 → total = 99

✔ Answer: 99.00

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Now let’s list all final answers clearly:

Final Answer:
1) 49.00
2) 93.60
3) 76.80
4) 91.00
5) 82.80
6) 61.50
7) 210.60
8) 120.00
9) 99.00