<Question>
An equiangular octagon has four sides of length $1$ and four sides of length $\frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?
</Question>

<think>
<Plan_1>
Okay, so I need to find the area of an equiangular octagon where the sides alternate between lengths 1 and √2/2. Hmm, equiangular octagon means all the internal angles are equal, right? But the sides alternate between two different lengths. Also, no two consecutive sides are the same, which makes sense since they alternate. 

First, I remember that in a regular octagon (both equiangular and equilateral), each internal angle is 135 degrees. Since this octagon is equiangular, each internal angle should also be 135 degrees. So even though the sides alternate, the angles are all the same. That might help in calculating the area.

Now, the sides alternate between 1 and √2/2. Let me confirm: four sides of 1 and four sides of √2/2, arranged so they don't repeat consecutively. So the sequence would be 1, √2/2, 1, √2/2, etc. , right? So the octagon has eight sides, alternating between these two lengths.

Since it's equiangular, maybe I can model it as a combination of shapes? Like, perhaps breaking it down into triangles or other polygons? Alternatively, maybe using coordinate geometry by placing the octagon on a coordinate system and calculating the coordinates of each vertex, then using the shoelace formula to find the area. That might be a systematic approach, especially since it's a polygon with equal angles but alternating side lengths.

Let me consider the coordinate system approach. If I can determine the coordinates of all eight vertices, I can apply the shoelace formula. To do that, I need to figure out the direction each side is pointing. Since all angles are 135 degrees, the direction changes by 45 degrees each time, right?

<prompt_1.1> Consider using coordinate geometry to place the octagon on a coordinate system and calculate the area using the shoelace formula. </prompt_1.1>

But how do I determine the radius R? Maybe by considering the side lengths. Wait, in a regular octagon, the side length is related to the radius. But here, the side lengths are not all equal. However, since the octagon is equiangular, maybe the radius can be determined such that the sides alternate between 1 and √2/2.

Alternatively, perhaps using vectors or complex numbers to model each side.

<prompt_1.2> Explore using vectors or complex numbers to model each side of the octagon. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Wait, in a regular octagon, each turn between sides is 45 degrees because the external angle is 360/8 = 45 degrees. So if this octagon is equiangular, each external angle is 45 degrees. Therefore, each time we move from one side to the next, the direction changes by 45 degrees.

But since the side lengths alternate, the octagon might not be symmetric in the same way as a regular octagon. Hmm, so starting at a point, say (0,0), and moving along the x-axis for length 1, then turning 45 degrees, then moving for length √2/2, then another 45 degrees, and so on. But I need to make sure that after eight sides, the octagon closes back on itself. That might require solving for the radius or some other parameter.
</execution_1.1>

<execution_1.2>
Let me try that.

Starting at the origin, let's say the first side is along the positive x-axis with length 1. So the first vertex is at (1, 0). The next side is of length √2/2, and since the internal angle is 135 degrees, the external angle (the turn between sides) would be 45 degrees. Wait, in a regular polygon, the external angle is 360/n, which for octagon is 45 degrees. So each time, we turn 45 degrees. But since the octagon is equiangular, the direction of each side is rotated by 45 degrees from the previous one.

But the side lengths alternate, so the lengths go 1, √2/2, 1, √2/2, etc. So each consecutive side is turned 45 degrees from the previous, but with alternating lengths.

Therefore, starting at (0,0), moving along the x-axis 1 unit to (1,0). Then turning 45 degrees upwards (since internal angle is 135, external angle is 45). Then moving √2/2 units in that direction. Then another 45 degrees turn (total 90 degrees from start), and moving 1 unit, etc. 

But to model this, I need to calculate each subsequent direction and the corresponding side length. Then sum up all the displacement vectors to ensure that the octagon closes back on itself. Because otherwise, the coordinates might not close properly.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Let me formalize this. Let's model each side as a vector in the plane. The direction of each vector is determined by the previous direction plus 45 degrees (π/4 radians). The magnitude alternates between 1 and √2/2. Since the octagon is closed, the sum of all vectors should be zero.

So, starting with the first vector along the x-axis: **v₁** = (1, 0). The next vector **v₂** has magnitude √2/2 and direction 45 degrees from the previous. Then **v₃** has magnitude 1 and direction 90 degrees from the starting x-axis, and so on.

Wait, but each subsequent vector is rotated by 45 degrees from the previous. So the angles of the vectors are 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and then back to 0°, right? But the magnitudes alternate between 1 and √2/2.

<prompt_2.1> Calculate the vectors for each side of the octagon and solve for the radius or parameters to ensure the polygon closes. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
But since the sides alternate, starting with 1, then √2/2, then 1, etc. , the magnitudes would be 1, √2/2, 1, √2/2, 1, √2/2, 1, √2/2. 

Therefore, the vectors would be:

**v₁** = 1*(cos 0°, sin 0°) = (1, 0)

**v₂** = (√2/2)*(cos 45°, sin 45°) = (√2/2 * √2/2, √2/2 * √2/2) = (0.5, 0.5)

**v₃** = 1*(cos 90°, sin 90°) = (0, 1)

**v₄** = (√2/2)*(cos 135°, sin 135°) = (√2/2 * -√2/2, √2/2 * √2/2) = (-0.5, 0.5)

**v₅** = 1*(cos 180°, sin 180°) = (-1, 0)

**v₆** = (√2/2)*(cos 225°, sin 225°) = (√2/2 * -√2/2, √2/2 * -√2/2) = (-0.5, -0.5)

**v₇** = 1*(cos 270°, sin 270°) = (0, -1)

**v₈** = (√2/2)*(cos 315°, sin 315°) = (√2/2 * √2/2, √2/2 * -√2/2) = (0.5, -0.5)

Then, summing all these vectors:

x-components: 1 + 0.5 + 0 - 0.5 -1 -0.5 +0 +0.5 = Let's compute step by step:

1 (from v₁)

+0.5 (v₂) = 1.5

+0 (v₃) = 1.5

-0.5 (v₄) = 1

-1 (v₅) = 0

-0.5 (v₆) = -0.5

+0 (v₇) = -0.5

+0.5 (v₈) = 0

Similarly, y-components:

0 (v₁)

+0.5 (v₂) = 0.5

+1 (v₃) = 1.5

+0.5 (v₄) = 2

+0 (v₅) = 2

-0.5 (v₆) = 1.5

-1 (v₇) = 0.5

-0.5 (v₈) = 0

So the total displacement is (0,0). Wait, that can't be right. If all the vectors sum to zero, then starting at (0,0), you end up back at (0,0). But in reality, the octagon should be a closed shape. However, according to these calculations, the sum of the vectors is zero.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
But that would mean that the octagon is closed. But wait, in reality, the octagon is not regular, so maybe this alternating of sides causes the vectors to cancel out? Hmm, but that seems counterintuitive. Let me check my calculations again.

<prompt_3.1> Verify the vector sum calculation to ensure the octagon is closed. </prompt_3.1>

But if the vectors sum to zero, then this is a valid closed octagon. So perhaps the coordinates can be calculated step by step.

<prompt_3.2> Calculate the coordinates of each vertex of the octagon. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Calculating x-components:

1 (from v₁)

+0.5 (v₂) = 1.5

+0 (v₃) = 1.5

-0.5 (v₄) = 1

-1 (v₅) = 0

-0.5 (v₆) = -0.5

+0 (v₇) = -0.5

+0.5 (v₈) = 0
</execution_3.1>

<execution_3.2>
Wait, but maybe I made a mistake in the directions. Let me confirm the angles. Each subsequent side is turned 45 degrees from the previous. But in a regular octagon, the direction of the sides would be at 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and then back to 360°=0°. So yes, each side is 45° apart. So the angles are correct.

But the side lengths alternate between 1 and √2/2. So starting with 1, then √2/2, then 1, etc. , as I did. Then the sum of the vectors is indeed zero. So that seems correct. Therefore, the octagon can be constructed by these vectors, and the area can be calculated using the shoelace formula.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
But how do I get the coordinates of each vertex? Let's compute them step by step.

<prompt_4.1> Use the shoelace formula to calculate the area of the octagon based on the calculated coordinates. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Starting at point P₀ = (0, 0).

After **v₁** (1, 0): P₁ = (1, 0)

After **v₂** (0.5, 0.5): P₂ = (1 + 0.5, 0 + 0.5) = (1.5, 0.5)

After **v₃** (0, 1): P₃ = (1.5 + 0, 0.5 + 1) = (1.5, 1.5)

After **v₄** (-0.5, 0.5): P₄ = (1.5 - 0.5, 1.5 + 0.5) = (1, 2)

After **v₅** (-1, 0): P₅ = (1 - 1, 2 + 0) = (0, 2)

After **v₆** (-0.5, -0.5): P₆ = (0 - 0.5, 2 - 0.5) = (-0.5, 1.5)

After **v₇** (0, -1): P₇ = (-0.5 + 0, 1.5 - 1) = (-0.5, 0.5)

After **v₈** (0.5, -0.5): P₈ = (-0.5 + 0.5, 0.5 - 0.5) = (0, 0)

So the coordinates of the octagon are:

P₀ = (0, 0)

P₁ = (1, 0)

P₂ = (1.5, 0.5)

P₃ = (1.5, 1.5)

P₄ = (1, 2)

P₅ = (0, 2)

P₆ = (-0.5, 1.5)

P₇ = (-0.5, 0.5)

P₈ = (0, 0)

Wait, but the last point should connect back to the first, but according to this, P₈ is (0,0), which is the same as P₀. So the octagon is defined by these eight vertices. Now, to compute the area using the shoelace formula, we can list the coordinates in order and apply the formula.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But let me verify the coordinates again. Let's retrace the steps:

<prompt_5.1> Verify the calculated coordinates of the octagon. </prompt_5.1>

Now, applying the shoelace formula. The formula is:

Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|

where (x_{n+1}, y_{n+1}) = (x_1, y_1).

So let's list the coordinates in order:

<prompt_5.2> Apply the shoelace formula to calculate the area of the octagon using the verified coordinates. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Starting at (0,0).

First side: length 1 along x-axis: ends at (1,0).

Second side: length √2/2 at 45 degrees: displacement (0.5, 0.5), so new position (1 + 0.5, 0 + 0.5) = (1.5, 0.5).

Third side: length 1 at 90 degrees (straight up): displacement (0,1), so (1.5 + 0, 0.5 + 1) = (1.5, 1.5).

Fourth side: length √2/2 at 135 degrees: displacement (-0.5, 0.5), so (1.5 - 0.5, 1.5 + 0.5) = (1, 2).

Fifth side: length 1 at 180 degrees: displacement (-1, 0), so (1 - 1, 2 + 0) = (0, 2).

Sixth side: length √2/2 at 225 degrees: displacement (-0.5, -0.5), so (0 - 0.5, 2 - 0.5) = (-0.5, 1.5).

Seventh side: length 1 at 270 degrees: displacement (0, -1), so (-0.5 + 0, 1.5 - 1) = (-0.5, 0.5).

Eighth side: length √2/2 at 315 degrees: displacement (0.5, -0.5), so (-0.5 + 0.5, 0.5 - 0.5) = (0, 0).
</execution_5.1>

<execution_5.2>
1. (0, 0)
2. (1, 0)
3. (1.5, 0.5)
4. (1.5, 1.5)
5. (1, 2)
6. (0, 2)
7. (-0.5, 1.5)
8. (-0.5, 0.5)
9. (0, 0)  // Closing the polygon

Now, compute each term x_i y_{i+1} - x_{i+1} y_i for i from 1 to 8.

Let's compute each term step by step:

Term 1: i=1 (0,0) to i+1=2 (1,0)
x1 y2 - x2 y1 = 0*0 - 1*0 = 0 - 0 = 0

Term 2: i=2 (1,0) to i+1=3 (1.5,0.5)
x2 y3 - x3 y2 = 1*0.5 - 1.5*0 = 0.5 - 0 = 0.5

Term 3: i=3 (1.5,0.5) to i+1=4 (1.5,1.5)
x3 y4 - x4 y3 = 1.5*1.5 - 1.5*0.5 = 2.25 - 0.75 = 1.5

Term 4: i=4 (1.5,1.5) to i+1=5 (1,2)
x4 y5 - x5 y4 = 1.5*2 - 1*1.5 = 3 - 1.5 = 1.5

Term 5: i=5 (1,2) to i+1=6 (0,2)
x5 y6 - x6 y5 = 1*2 - 0*2 = 2 - 0 = 2

Term 6: i=6 (0,2) to i+1=7 (-0.5,1.5)
x6 y7 - x7 y6 = 0*1.5 - (-0.5)*2 = 0 + 1 = 1

Term 7: i=7 (-0.5,1.5) to i+1=8 (-0.5,0.5)
x7 y8 - x8 y7 = (-0.5)*0.5 - (-0.5)*1.5 = (-0.25) + 0.75 = 0.5

Term 8: i=8 (-0.5,0.5) to i+1=9 (0,0)
x8 y9 - x9 y8 = (-0.5)*0 - 0*0.5 = 0 - 0 = 0

Adding all these terms: 0 + 0.5 + 1.5 + 1.5 + 2 + 1 + 0.5 + 0 = 7

Therefore, the area is 1/2 * |7| = 3.5. So 3.5 square units.
</execution_5.2>
</Execution_5>

<Plan_6>
Based on execution_5.2:
Wait, 3.5 is 7/2. But let me check the calculations again because that seems a bit low. Let me verify each term:

<prompt_6.1> Re-evaluate the shoelace formula calculations to verify the area. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
Term 1: 0, correct.

Term 2: 1*0.5 - 1.5*0 = 0.5, correct.

Term 3: 1.5*1.5 = 2.25, 1.5*0.5 = 0.75, 2.25 - 0.75 = 1.5, correct.

Term 4: 1.5*2 = 3, 1*1.5 = 1.5, 3 - 1.5 = 1.5, correct.

Term 5: 1*2 = 2, 0*2 = 0, 2 - 0 = 2, correct.

Term 6: 0*1.5 = 0, -0.5*2 = -1, but since it's x6 y7 - x7 y6, which is 0*1.5 - (-0.5)*2 = 0 + 1 = 1, correct.

Term 7: (-0.5)*0.5 = -0.25, (-0.5)*1.5 = -0.75, so x7 y8 - x8 y7 = (-0.5)*0.5 - (-0.5)*1.5 = (-0.25) - (-0.75) = 0.5, correct.

Term 8: 0, correct.

Sum: 0 + 0.5 + 1.5 + 1.5 + 2 + 1 + 0.5 + 0 = 0.5 + 1.5 = 2; 2 + 1.5 = 3.5; 3.5 + 2 = 5.5; 5.5 +1 =6.5; 6.5 +0.5=7; 7 +0=7.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
So the area is 7/2 = 3.5. But let me think if this is correct. However, I need to check if the coordinates are correctly ordered and if the polygon is simple (non-intersecting).

<prompt_7.1> Verify the shoelace formula results and consider the symmetry of the octagon. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
Looking at the coordinates:

Starting at (0,0), going to (1,0), then (1.5,0.5), (1.5,1.5), (1,2), (0,2), (-0.5,1.5), (-0.5,0.5), (-0.5,0.5) to (0,0). Hmm, actually, when connecting these points, does the octagon intersect itself? Let me visualize:

From (0,0) to (1,0) to (1.5,0.5) to (1.5,1.5) to (1,2) to (0,2) to (-0.5,1.5) to (-0.5,0.5) back to (0,0). It seems like a convex octagon? Wait, but (-0.5,0.5) to (0,0) would cross over previous sides? Wait, no, because all the turns are 45 degrees, so each subsequent side is turning left (assuming counterclockwise order). Wait, but did I maintain the counterclockwise order in the coordinates? Let me check:

Starting at (0,0), moving to (1,0) (right), then to (1.5,0.5) (up and right), then to (1.5,1.5) (up), then to (1,2) (left and up), then to (0,2) (left), then to (-0.5,1.5) (left and down), then to (-0.5,0.5) (down), then back to (0,0). This seems to be a counterclockwise traversal. So the polygon is convex? But in a regular octagon, all vertices are convex. However, in this case, the sides alternate between lengths 1 and √2/2, so maybe it's not regular but still convex. 

But the shoelace formula gives 3.5. However, I need to confirm if the coordinates are correctly ordered and non-intersecting. Given the way the sides are constructed, each subsequent vertex is obtained by adding the next vector, which is a rotation of 45 degrees and the specified length. Since the vectors sum to zero, the polygon is closed. However, the shape might have overlapping edges? Wait, no, because each turn is 45 degrees, so each subsequent edge is at a 45-degree angle from the previous, but the lengths alternate. However, the calculated coordinates result in a simple, convex octagon.
</execution_7.1>
</Execution_7>

<Plan_8>
Based on execution_7.1:
But let me cross-verify with another approach. Maybe dividing the octagon into simpler shapes like rectangles and triangles.

Looking at the coordinates:

<prompt_8.1> Cross-verify the shoelace formula results by dividing the octagon into simpler shapes. </prompt_8.1>
</Plan_8>

<Execution_8>
<execution_8.1>
The octagon has a vertical line of symmetry along x=0.5? Wait, looking at the coordinates, the octagon is symmetric with respect to the y-axis? Wait, no. Let's check:

Points:

(0,0), (1,0), (1.5,0.5), (1.5,1.5), (1,2), (0,2), (-0.5,1.5), (-0.5,0.5), (0,0). 

If we reflect over the y-axis (x=0), the point (1,0) would map to (-1,0), which isn't in the list. So maybe symmetry is different. Alternatively, maybe rotational symmetry. But perhaps it's better not to get bogged down in symmetry unless it helps.

Alternatively, perhaps the area can be calculated by breaking the octagon into triangles or other polygons. For example, using the coordinates, we can divide the octagon into triangles from the origin, but since the octagon isn't star-shaped, that might not work. Alternatively, divide into trapezoids or triangles between consecutive points.

Alternatively, using the shoelace formula is straightforward if the coordinates are correct.
</execution_8.1>
</Execution_8>

<Plan_9>
Based on execution_8.1:
But let me check once again. Alternatively, maybe there's a mistake in the order of the coordinates. Because if the octagon is constructed by alternating sides of 1 and √2/2, the order of the vertices might not be as I listed them. Wait, perhaps the order is different. Let me think again about how the octagon is constructed.

<prompt_9.1> Re-evaluate the construction of the octagon and verify the coordinate order. </prompt_9.1>
</Plan_9>

<Execution_9>
<execution_9.1>
Starting at (0,0), moving right 1 unit to (1,0). Then, at 45 degrees, moving √2/2 units to (1.5,0.5). Then up 1 unit to (1.5,1.5). Then at 135 degrees from the previous direction (which was up), so 135 degrees from the positive x-axis? Wait, no. Each turn is 45 degrees, but the direction of each side is determined by the previous direction plus 45 degrees. Wait, actually, in a regular octagon, each subsequent side is rotated by 45 degrees from the previous. But in this case, since the side lengths alternate, perhaps the direction changes by 45 degrees each time, regardless of the side length. 

Wait, but regardless of the side length, the external angle is 45 degrees. So each consecutive side is turned 45 degrees from the previous one. Therefore, the direction of each side is previous direction plus 45 degrees. So starting along the x-axis (0 degrees), next is 45 degrees, then 90 degrees, etc. , as I did before. So the angles of the sides are 0°,45°,90°,135°,180°,225°,270°,315°, and back to 360°=0°. 

Therefore, the order of the coordinates as calculated seems correct. Therefore, the shoelace formula applied to these coordinates gives 3.5. But is that accurate? Let me cross-verify by calculating the area step by step again.

Alternatively, perhaps I miscalculated the shoelace terms. Let me recompute each term:

Term 1: x1=0, y2=0; x2=1, y1=0. So 0*0 - 1*0 = 0.

Term 2: x2=1, y3=0.5; x3=1.5, y2=0. So 1*0.5 - 1.5*0 = 0.5.

Term 3: x3=1.5, y4=1.5; x4=1.5, y3=0.5. So 1.5*1.5 - 1.5*0.5 = 2.25 - 0.75 = 1.5.

Term 4: x4=1.5, y5=2; x5=1, y4=1.5. So 1.5*2 - 1*1.5 = 3 - 1.5 = 1.5.

Term 5: x5=1, y6=2; x6=0, y5=2. So 1*2 - 0*2 = 2 - 0 = 2.

Term 6: x6=0, y7=1.5; x7=-0.5, y6=2. So 0*1.5 - (-0.5)*2 = 0 + 1 = 1.

Term 7: x7=-0.5, y8=0.5; x8=-0.5, y7=1.5. So (-0.5)*0.5 - (-0.5)*1.5 = (-0.25) - (-0.75) = 0.5.

Term 8: x8=-0.5, y9=0; x9=0, y8=0.5. So (-0.5)*0 - 0*0.5 = 0 - 0 = 0.

Adding up: 0 + 0.5 +1.5 +1.5 +2 +1 +0.5 +0 = 7. So 7/2 = 3.5.
</execution_9.1>
</Execution_9>

<Plan_10>
Based on execution_9.1:
But let me think, is there another way to compute the area? For example, using the formula for equiangular polygons. I recall that for a convex polygon with equal sides, there's a formula, but here sides are not equal. However, since the sides alternate, maybe there's a pattern.

Alternatively, perhaps using complex numbers to represent the vertices and compute the area. But that might be more complicated.

Alternatively, since all the sides are either length 1 or √2/2, and angles between them are 45 degrees, maybe each pair of sides (1 and √2/2) form a rhombus or something, but given the alternating lengths, perhaps each "unit" of the octagon can be considered as a combination of a square and triangles.

Wait, another approach: since the octagon is equiangular, all sides can be considered as vectors with angles 45 degrees apart, as we did before. The area can be calculated by summing up the areas of the triangles formed with the center. However, since the octagon is not regular, the distance from the center to each side (the radius) would vary. Therefore, this approach might not be straightforward.

Alternatively, using the formula for the area of a polygon with given side lengths and angles. But I don't recall a specific formula for an equiangular octagon with alternating sides. The alternative is to use the general formula for the area of a polygon with given side lengths and angles, but that might be complex.

Alternatively, since the octagon is determined by its side vectors, which we have already computed, and the shoelace formula gives 3.5, which is 7/2, and after cross-verifying the coordinates, perhaps that is correct. However, 3.5 seems a bit low. Let me check if there's a miscalculation in the coordinates.

<prompt_10.1> Perform a final verification of the shoelace formula and coordinate calculations to ensure accuracy. </prompt_10.1>
</Plan_10>

<Execution_10>
<execution_10.1>
Wait, let's recompute the coordinates step by step once again:

Starting at (0,0).

1. Move right 1 unit: (1,0).

2. Turn 45 degrees (upwards), move √2/2 units: displacement (cos45, sin45) = (√2/2 * √2/2, √2/2 * √2/2) = (0.5, 0.5). So new position: (1 + 0.5, 0 + 0.5) = (1.5, 0.5).

3. Turn another 45 degrees (total 90 degrees from start), move 1 unit: displacement (0,1). New position: (1.5, 0.5 + 1) = (1.5, 1.5).

4. Turn another 45 degrees (total 135 degrees), move √2/2 units: displacement (cos135°, sin135°) = (-√2/2, √2/2). So displacement (-0.5, 0.5). New position: (1.5 - 0.5, 1.5 + 0.5) = (1, 2).

5. Turn another 45 degrees (total 180 degrees), move 1 unit: displacement (-1, 0). New position: (1 - 1, 2 + 0) = (0, 2).

6. Turn another 45 degrees (total 225 degrees), move √2/2 units: displacement (cos225°, sin225°) = (-√2/2, -√2/2). So displacement (-0.5, -0.5). New position: (0 - 0.5, 2 - 0.5) = (-0.5, 1.5).

7. Turn another 45 degrees (total 270 degrees), move 1 unit: displacement (0, -1). New position: (-0.5, 1.5 - 1) = (-0.5, 0.5).

8. Turn another 45 degrees (total 315 degrees), move √2/2 units: displacement (cos315°, sin315°) = (√2/2, -√2/2). So displacement (0.5, -0.5). New position: (-0.5 + 0.5, 0.5 - 0.5) = (0, 0).

All coordinates check out. Shoelace formula steps also seem correct. Therefore, the area is indeed 3.5, which is 7/2. 

But let me check if there's a different approach. For example, maybe dividing the octagon into simpler components. Let's see, the octagon can be divided into a central square and four rectangles and four triangles? Wait, given the coordinates, perhaps not. Alternatively, since the octagon is symmetric in some way, maybe we can compute the area by calculating the area of a section and multiplying.

Looking at the coordinates, from (0,0) to (1,0) to (1.5,0.5) to (1.5,1.5) to (1,2) to (0,2) to (-0.5,1.5) to (-0.5,0.5) to (0,0). If we consider the right half of the octagon (from x=0 to x=1.5), it has a sort of trapezoid shape. Similarly, the left half is a mirror image.

Alternatively, compute the area of the right half and double it. The right half would be from x=0 to x=1.5, bounded by the points (0,0), (1,0), (1.5,0.5), (1.5,1.5), (1,2), (0,2), (-0.5,1.5), (-0.5,0.5), but that seems complicated. Alternatively, use the shoelace formula on the right half.

But perhaps not. Given the time I've spent and the checks I've done, I think the shoelace formula is the most reliable here. Therefore, the area is 7/2.
</execution_10.1>
</Execution_10>

<Plan_11>
Based on execution_10.1:
**Final Answer**
The area of the octagon is \boxed{\dfrac{7}{2}}.
</Plan_11>

<Execution_11>
No prompt to execute.
</Execution_11>

<Final_answer>
Therefore, the area of the octagon is $\boxed{\frac{7}{2}}$.
</Final_answer>
</think>