Explanation:
Let's solve each problem step by step. We need to identify the polygon, calculate its area and perimeter, and fill in the blanks.
The formula for the area of a regular polygon is:
Area = (1/2) × Perimeter × apothem
Or, more commonly:
Area = (1/2) × (number of sides × side length) × apothem
So, Area = (1/2) × n × s × a
The perimeter is:
Perimeter = n × s
We will use these formulas for each problem.
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1) s = 6.8 mm, a = 3.23 mm
- The shape has 5 sides → Pentagon
- n = 5
- Perimeter = 5 × 6.8 = 34.0 mm
- Area = (1/2) × 5 × 6.8 × 3.23
= (1/2) × 5 × 6.8 × 3.23
= 2.5 × 6.8 × 3.23
= 17 × 3.23 = 54.91 mm²
- Type: Pentagon
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2) s = 5.1 ft, a = 2.43 ft
- The shape has 5 sides → Pentagon
- n = 5
- Perimeter = 5 × 5.1 = 25.5 ft
- Area = (1/2) × 5 × 5.1 × 2.43
= 2.5 × 5.1 × 2.43
= 12.75 × 2.43 = 30.9525 ft²
- Type: Pentagon
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3) s = 3.3 ft, a = 1.65 ft
- The shape has 8 sides → Octagon
- n = 8
- Perimeter = 8 × 3.3 = 26.4 ft
- Area = (1/2) × 8 × 3.3 × 1.65
= 4 × 3.3 × 1.65
= 13.2 × 1.65 = 21.78 ft²
- Type: Octagon
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4) s = 3.1 yds, a = 1.55 yds
- The shape has 6 sides → Hexagon
- n = 6
- Perimeter = 6 × 3.1 = 18.6 yds
- Area = (1/2) × 6 × 3.1 × 1.55
= 3 × 3.1 × 1.55
= 9.3 × 1.55 = 14.415 yds²
- Type: Hexagon
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5) s = 2.7 inches, a = 1.35 inches
- The shape has 8 sides → Octagon
- n = 8
- Perimeter = 8 × 2.7 = 21.6 inches
- Area = (1/2) × 8 × 2.7 × 1.35
= 4 × 2.7 × 1.35
= 10.8 × 1.35 = 14.58 in²
- Type: Octagon
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6) s = 6 mm, a = 2.6 mm
- The shape has 8 sides → Octagon
- n = 8
- Perimeter = 8 × 6 = 48 mm
- Area = (1/2) × 8 × 6 × 2.6
= 4 × 6 × 2.6
= 24 × 2.6 = 62.4 mm²
- Type: Octagon
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7) s = 5 yds, a = 2.5 yds
- The shape has 6 sides → Hexagon
- n = 6
- Perimeter = 6 × 5 = 30 yds
- Area = (1/2) × 6 × 5 × 2.5
= 3 × 5 × 2.5
= 15 × 2.5 = 37.5 yds²
- Type: Hexagon
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8) s = 2.4 cm, a = 1.2 cm
- The shape has 8 sides → Octagon
- n = 8
- Perimeter = 8 × 2.4 = 19.2 cm
- Area = (1/2) × 8 × 2.4 × 1.2
= 4 × 2.4 × 1.2
= 9.6 × 1.2 = 11.52 cm²
- Type: Octagon
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9) s = 6.7 inches, a = 3.35 inches
- The shape has 6 sides → Hexagon
- n = 6
- Perimeter = 6 × 6.7 = 40.2 inches
- Area = (1/2) × 6 × 6.7 × 3.35
= 3 × 6.7 × 3.35
= 20.1 × 3.35 = 67.335 in²
- Type: Hexagon
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Final Answer:
1) Type: Pentagon, Area: 54.91 mm², Perimeter: 34.0 mm
2) Type: Pentagon, Area: 30.9525 ft², Perimeter: 25.5 ft
3) Type: Octagon, Area: 21.78 ft², Perimeter: 26.4 ft
4) Type: Hexagon, Area: 14.415 yds², Perimeter: 18.6 yds
5) Type: Octagon, Area: 14.58 in², Perimeter: 21.6 in
6) Type: Octagon, Area: 62.4 mm², Perimeter: 48 mm
7) Type: Hexagon, Area: 37.5 yds², Perimeter: 30 yds
8) Type: Octagon, Area: 11.52 cm², Perimeter: 19.2 cm
9) Type: Hexagon, Area: 67.335 in², Perimeter: 40.2 in
Parent Tip: Review the logic above to help your child master the concept of polygons worksheet 5th grade.
