Question

7)
8
8.8
8)
19.8
23.9
9)
9.2
13.4
10)
6.5
9.4
11)
7.4
6.1
12)
13.1
12.6

Explanation:

To solve for the area of a regular polygon, we use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]
(where the apothem is the distance from the center to the midpoint of a side, and the perimeter is the sum of all side lengths).

Problem 7: Regular Pentagon

• Side length (\( s \)) = 8
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 8.8

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 8 = 40 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 40 \times 8.8 = 176 \]

Problem 8: Regular Octagon

• Side length (\( s \)) = 19.8
• Number of sides (\( n \)) = 8
• Apothem (\( a \)) = 23.9

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 8 \times 19.8 = 158.4 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 158.4 \times 23.9 = 1902.48 \]

Problem 9: Regular Pentagon

• Side length (\( s \)) = 13.4
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 9.2

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 13.4 = 67 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 67 \times 9.2 = 312.2 \]

Problem 10: Regular Pentagon

• Side length (\( s \)) = 9.4
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 6.5

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 9.4 = 47 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 47 \times 6.5 = 152.75 \]

Problem 11: Regular Octagon

• Side length (\( s \)) = 6.1
• Number of sides (\( n \)) = 8
• Apothem (\( a \)) = 7.4

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 8 \times 6.1 = 48.8 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 48.8 \times 7.4 = 180.64 \]

Problem 12: Regular Hexagon

• Side length (\( s \)) = 13.1
• Number of sides (\( n \)) = 6
• Apothem (\( a \)) = 12.6 (distance from center to side, equal to \( \frac{\sqrt{3}}{2} \times \text{side length} \) for a regular hexagon, but given directly here)

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 6 \times 13.1 = 78.6 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 78.6 \times 12.6 = 499.14 \]

Final Answers:

1. \(\boldsymbol{176}\)
2. \(\boldsymbol{1902.48}\)
3. \(\boldsymbol{312.2}\)
4. \(\boldsymbol{152.75}\)
5. \(\boldsymbol{180.64}\)
6. \(\boldsymbol{499.14}\)

Answer:

To solve for the area of a regular polygon, we use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]
(where the apothem is the distance from the center to the midpoint of a side, and the perimeter is the sum of all side lengths).

Problem 7: Regular Pentagon

• Side length (\( s \)) = 8
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 8.8

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 8 = 40 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 40 \times 8.8 = 176 \]

Problem 8: Regular Octagon

• Side length (\( s \)) = 19.8
• Number of sides (\( n \)) = 8
• Apothem (\( a \)) = 23.9

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 8 \times 19.8 = 158.4 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 158.4 \times 23.9 = 1902.48 \]

Problem 9: Regular Pentagon

• Side length (\( s \)) = 13.4
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 9.2

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 13.4 = 67 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 67 \times 9.2 = 312.2 \]

Problem 10: Regular Pentagon

• Side length (\( s \)) = 9.4
• Number of sides (\( n \)) = 5
• Apothem (\( a \)) = 6.5

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 5 \times 9.4 = 47 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 47 \times 6.5 = 152.75 \]

Problem 11: Regular Octagon

• Side length (\( s \)) = 6.1
• Number of sides (\( n \)) = 8
• Apothem (\( a \)) = 7.4

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 8 \times 6.1 = 48.8 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 48.8 \times 7.4 = 180.64 \]

Problem 12: Regular Hexagon

• Side length (\( s \)) = 13.1
• Number of sides (\( n \)) = 6
• Apothem (\( a \)) = 12.6 (distance from center to side, equal to \( \frac{\sqrt{3}}{2} \times \text{side length} \) for a regular hexagon, but given directly here)

Step 1: Calculate Perimeter

Perimeter (\( P \)) = \( n \times s = 6 \times 13.1 = 78.6 \)

Step 2: Calculate Area

\[ \text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 78.6 \times 12.6 = 499.14 \]

Final Answers:

1. \(\boldsymbol{176}\)
2. \(\boldsymbol{1902.48}\)
3. \(\boldsymbol{312.2}\)
4. \(\boldsymbol{152.75}\)
5. \(\boldsymbol{180.64}\)
6. \(\boldsymbol{499.14}\)