Answer:
The formula for the area of a regular octagon is:
A = 2(1 + √2) × apothem^2
So, we can use this formula and the given values to find the length of each side of the octagon:
288 = 2(1 + √2) × 12^2
Dividing both sides by 2(1 + √2), we get:
12^2 = 288 ÷ 2(1 + √2)
Simplifying the right-hand side, we get:
12^2 = 72(1 - √2)
Dividing both sides by 72(1 - √2), we get:
12^2 ÷ 72(1 - √2) = 1
Simplifying the left-hand side, we get:
2 - √2 = 1
Adding √2 to both sides, we get:
2 = 1 + √2
This is not true, so there must be an error in our calculations.
Checking our work, we find that the formula we used for the area of a regular octagon is incorrect. The correct formula is:
A = 2 × (1 + √2) × apothem × side
Using this formula and the given values, we can solve for the length of each side of the octagon:
288 = 2 × (1 + √2) × 12 × side
Dividing both sides by 2 × (1 + √2) × 12, we get:
side = 288 ÷ (2 × (1 + √2) × 12)
Simplifying the right-hand side, we get:
side = 6(√2 - 1)
Therefore, the length of each side of the octagon is approximately 4.83 cm (rounded to two decimal places).