Let’s solve each polygon one by one. We’ll find the
perimeter first (add up all the sides), then the
area using the formula:
>
Area = ½ × Perimeter × Apothem
*(The apothem is the distance from the center to the middle of a side — shown as “a” or with a perpendicular line in the diagrams.)*
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🔷 Top Left: Regular Pentagon
- Side length = 18 units
- Number of sides = 5 → so perimeter = 5 × 18 =
90 units
- Apothem = 15 units
- Area = ½ × 90 × 15 = 45 × 15 =
675 square units
✔ Check: 45 × 15 → 40×15=600, 5×15=75 → 600+75=675 ✔️
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🔷 Top Right: Regular Octagon
- Side length = 10 m
- Number of sides = 8 → perimeter = 8 × 10 =
80 m
- Apothem = 12.1 m
- Area = ½ × 80 × 12.1 = 40 × 12.1
Let’s compute 40 × 12.1:
→ 40 × 12 = 480
→ 40 × 0.1 = 4
→ Total = 480 + 4 =
484 m²
✔ Check: 40 × 12.1 = 484 ✔️
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🔷 Middle Left: Regular Octagon
- Side length = 5 cm
- Number of sides = 8 → perimeter = 8 × 5 =
40 cm
- Apothem = 6 cm
- Area = ½ × 40 × 6 = 20 × 6 =
120 cm²
✔ Simple multiplication ✔️
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🔷 Middle Right: Regular Hexagon
- Side length = 5 cm
- Number of sides = 6 → perimeter = 6 × 5 =
30 cm
- Apothem = 4.3 cm
- Area = ½ × 30 × 4.3 = 15 × 4.3
Compute 15 × 4.3:
→ 15 × 4 = 60
→ 15 × 0.3 = 4.5
→ Total = 60 + 4.5 =
64.5 cm²
✔ Check: 15 × 4.3 = 64.5 ✔️
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🔷 Bottom Left: Regular Pentagon
- Side length = 10 cm
- Number of sides = 5 → perimeter = 5 × 10 =
50 cm
- Apothem = 6.9 cm
- Area = ½ × 50 × 6.9 = 25 × 6.9
Compute 25 × 6.9:
→ 25 × 7 = 175
→ But we did 25 × 0.1 too much → subtract 25 × 0.1 = 2.5
→ So 175 - 2.5 =
172.5 cm²
Alternatively:
25 × 6 = 150
25 × 0.9 = 22.5
Total = 150 + 22.5 =
172.5 cm² ✔️
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🔷 Bottom Right: Regular Pentagon
- Side length = 2 cm
- Number of sides = 5 → perimeter = 5 × 2 =
10 cm
- Apothem = 1.4 cm
- Area = ½ × 10 × 1.4 = 5 × 1.4 =
7 cm²
✔ 5 × 1.4 = 7 ✔️
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Final Answer:
Top Left (Pentagon):
Perimeter: 90 units
Area: 675 square units
Top Right (Octagon):
Perimeter: 80 m
Area: 484 m²
Middle Left (Octagon):
Perimeter: 40 cm
Area: 120 cm²
Middle Right (Hexagon):
Perimeter: 30 cm
Area: 64.5 cm²
Bottom Left (Pentagon):
Perimeter: 50 cm
Area: 172.5 cm²
Bottom Right (Pentagon):
Perimeter: 10 cm
Area: 7 cm²
Parent Tip: Review the logic above to help your child master the concept of perimeter of a polygon worksheet.
