We have a total of 5 small squares with side of 1 and 4 small right isosceles triangle with diagonal of 2^(1/2) or side of 1
--> Area = 5*1*1 + 4*1/2*1*1 = 7
--> A

Here,

Let us assume the polygon to be a square, with 4 △'s as in the diagram attached

So the area of the polygon = Area of the square - 4 * Area of the △

Now let us take any one △,

We can see the hypotenuse = \(\sqrt2\)and it is a 45 - 45 - 90 triangle

Therefore each side is of equal length = 1

Now the area of the △ = \(\frac{1}{2} * 1 * 1 = \frac{1}{2}\)

Area of the square = \(side^2 = 3^2 = 9\)

Therefore the area of the polygon = Area of the square - 4 * Area of the △ = \(9 - (4 * \frac{1}{2}) = 7\)

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Basically just imagine that you have small squares and half squares and you are good to go

The numbers in the green are the areas of the shapes.

Process
1. Inscribe the octagon in a square.

2. At the four corners of the square, there will be four triangles. These are formed by the gap between the perimeter of the octagon and the perimeter of the square.

3. Those four triangles are isosceles due to the fact that the octagon is equiangular:
- The sum of the degrees of the internal angles of an octagon is 180(6) = 1080 degrees, meaning that, since this is an equiangular octagon, we have eight internal angles measuring 135 degrees.
- The sides of the square overlap with the sides of the octagon such that each internal angle of the octagon and an adjacent angle from one of the triangles are supplementary (add up to 180 degrees).
- Because the internal angles are all equal, these angles within the triangles will also be equal, creating isosceles right triangles.

4. Because the four triangles are isosceles and have a sqrt(2) hypotenuse, their legs both measure 1.

5. Because their legs measure 1, the length of one side of the square is 3 and the area of the square is 9.

6. Because their legs measure 1, the area of the triangles is 1/2 apiece.

7. To find the area of the octagon, subtract the area of the four triangles from the area of the square:
9 - (1/2)(4) = 9 - 2 = 7.

Hence, A is our answer.

I apologize for not uploading any pictures!

this is only possible, if every angle measures as (90+45)
there 4 right triangles with legs 1 and 5 squares with side 1
summing up 4*(1*1/2) + 5*(1*1) = 2+5 = 7

Ah, I think I see what you're saying.

The problem did say that the figure was equiangular, though. Even if the sides of an octagon are different lengths, the interior angles should all be the same if it's equiangular.

For one angle to be more or less than 135, another angle would have to adjust to compensate, and then it wouldn't be equiangular.

Am I understanding that right?

I did exactly mean that. If it were not for equiangular property, this solution would be missing something
firstly, define each angle and then strategize
(n-2)*180/n, where n=8 is the implied condition here for (90+45)

Cool; so there's no flaw, then?