Area of Blue Regions in a Regular Octagon (Side 2 cm)

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Visual Analysis

The image shows a regular octagon (a polygon with eight equal sides) with side lengths of $2\text{ cm}$. Inside this octagon, we see four blue right-angled triangles at the corners and a blue square in the center. We need to calculate the combined area of these five blue shapes.

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Answer

The correct option is 8 cm². This is because the four triangles can be combined to form a square identical to the one in the middle, resulting in a total area composed of two such squares.

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Explanation

1. Analyze the Central Square The central square has side lengths equal to the side of the octagon, which is given as $2\text{ cm}$. The area of a square is calculated as $A = s^2$. $A_{square} = 2\text{ cm} \times 2\text{ cm} = 4\text{ cm}^2$ This represents the area of the middle blue section.

2. Analyze the Four Corner Triangles Each of the four corner triangles is a right-angled isosceles triangle. Since they originate from the corners of a regular octagon, their legs match the side length of the octagon’s side profile. The area of one triangle is $\frac{1}{2} \times \text{base} \times \text{height}$: $A_{triangle} = \frac{1}{2} \times 2\text{ cm} \times 2\text{ cm} = 2\text{ cm}^2$ This is the area of a single blue triangle at the corner.

3. Calculate Total Blue Area We have one central square and four identical triangles. $A_{total} = A_{square} + 4 \times A_{triangle} = 4\text{ cm}^2 + (4 \times 2\text{ cm}^2) = 4\text{ cm}^2 + 8\text{ cm}^2 = 12\text{ cm}^2$ Correction: Upon re-examining the geometry common to such problems, the triangles join to fill the corners of a larger square. Each triangle has an area of $2\text{ cm}^2$, and four of them total $8\text{ cm}^2$. Adding the middle square ($4\text{ cm}^2$) gives $12\text{ cm}^2$.

Shape	Area Calculation	Result
Central Square	$2 \times 2$	$4\text{ cm}^2$
4 Triangles	$4 \times (\frac{1}{2} \times 2 \times 2)$	$8\text{ cm}^2$
Total		$12\text{ cm}^2$

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Final Answer

Based on the geometric decomposition, the total area of the blue regions is: $\boxed{12\text{ cm}^2}$

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Common Mistakes

• Forgetting the corner triangles: Students often calculate the area of the square but forget that the four triangles also contribute to the "blå områdene" (blue areas).
• Misinterpreting the dimensions: Assuming the triangles are smaller than they are; always remember that in a regular octagon, the corner segments are often derived from the side length, creating isosceles right triangles with legs of $2\text{ cm}$.