Two unit squares share the same center. The overlapping region of the two squares is an octagon with perimeter 3.5. What is the area of the octagon?

I don't know how to do this. Could you please help me?

I will assume the octagon is regular, so each side is 3.5/8 = 7/16.

Now just divide the octagon into eight isosceles triangles, each with base 7/16 and vertex angle 45°.

The area of each triangle is easy to do, and just multiply it by 8.

Sure! Let's break down the problem step by step.

First, let's consider the two unit squares sharing the same center. A unit square has sides of length 1, so the diagonal of each square would be √2 using the Pythagorean theorem.

To find the perimeter of the octagon, we need to determine the length of its sides. Since the octagon is created by the overlapping region of the squares, we can see that four of the sides of the octagon are parts of the sides of the squares, with a length of 1.

Now, if we draw a line from the center of the squares to the midpoint of one of the sides, it will pass through the vertex of the octagon. This line is the radius of a circle inscribed within the octagon, which is also equal to the apothem (the distance from the center to the midpoint of a side) of the octagon.

Since the diagonal of a square forms a right triangle with the side of the square, we can use the Pythagorean theorem to find the length of this radius/apothem. The hypotenuse is √2 (the diagonal of the square), and since the octagon is made up of isosceles triangles (each having two sides of length 1 and the third side as the radius), the base is 1. Therefore, the height (radius) can be found using the Pythagorean theorem as √(√2⁻² - 1⁻²).

Now, to find the remaining four sides of the octagon, we can use the formula for the circumference of a circle (2πr), where r is the radius we just found. Since the perimeter of the octagon is given as 3.5, we can subtract the length of the four square sides (4) from the octagon's perimeter to find the sum of the remaining four sides.

Finally, we can calculate the area of the octagon using the formula (1/2) * apothem * perimeter.

Let's do the calculations:

1. Find the radius/apothem:
radius = √(√2⁻² - 1⁻²) ≈ 0.316

2. Calculate the sum of the remaining four sides of the octagon:
perimeter of the octagon = 3.5
remaining sides = 3.5 - (4 * 1) = 3.5 - 4 = 0.5

3. Calculate the area of the octagon:
area = (1/2) * apothem * perimeter
= (1/2) * 0.316 * 0.5
≈ 0.079 square units

Therefore, the area of the octagon is approximately 0.079 square units.