To calculate the area of a regular octagon, there is a specific formula that can be used:
[tex]\[ \text{Area} = 2 \cdot (1 + \sqrt{2}) \cdot s^2 \][/tex]
Where [tex]\( s \)[/tex] is the side length of the octagon.
For a standard stop sign with a side length of 30 inches:
1. Identify the side length ([tex]\( s \)[/tex]) of the octagon, which is given as 30 inches.
2. Use the formula for the area of a regular octagon:
[tex]\[ \text{Area} = 2 \cdot (1 + \sqrt{2}) \cdot s^2 \][/tex]
3. Substitute the side length ([tex]\( s = 30 \)[/tex] inches) into the formula:
[tex]\[ \text{Area} = 2 \cdot (1 + \sqrt{2}) \cdot (30)^2 \][/tex]
4. Perform the calculations inside the parentheses first:
[tex]\[ 1 + \sqrt{2} \approx 2.414 \][/tex]
5. Multiply this value by 2:
[tex]\[ 2 \cdot 2.414 \approx 4.828 \][/tex]
6. Square the side length:
[tex]\[ (30)^2 = 900 \][/tex]
7. Finally, multiply these values together to find the area:
[tex]\[ \text{Area} \approx 4.828 \cdot 900 \approx 4345.584 \][/tex]
Therefore, the area of each stop sign is approximately 4345.584 square inches.
