Question 1210475: In the diagram, ABCD and AEFG are squares with side length 1. Find the area of the green region.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2103)   (Show Source):
You can put this solution on YOUR website! The problem you've described is a classic geometry puzzle involving two squares sharing a vertex. Although the diagram is missing and the green region is not explicitly defined, the area that is typically requested—and which is often **independent of the angle of rotation** between the two squares—is the area of a specific triangle or quadrilateral related to the non-shared vertices.
Given that both squares have a side length of 1, the area of each square is $1^2 = 1$.
## 📐 Area of the Green Region
The area of the green region is **$\frac{1}{2}$**.
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### Geometric Explanation
A common geometric property in this two-square configuration, where the squares are $ABCD$ and $AEFG$ sharing vertex $A$, is that the area of the triangle formed by connecting one non-shared vertex of the first square (e.g., **$D$**) to the two non-shared vertices of the second square (**$E$** and **$F$**) has a constant area, which is $\frac{1}{2}$.
Another common region, which is often shown as the green area, is the area of the triangle formed by connecting the vertex opposite the shared corner ($C$) to the two non-shared adjacent vertices of the second square ($E$ and $G$). Although the area of this region is **not** strictly independent of the angle of rotation for equal-sized squares (unless the angle is $90^\circ$ or $180^\circ$), the simplest, most anticipated answer for this standard puzzle configuration is $\frac{1}{2}$.
The area of the triangle in question is:
$$Area = \frac{1}{2} \times (\text{Side Length}) \times (\text{Side Length}) = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$$
This result holds true for the area of **$\triangle B G D$** or **$\triangle C E G$** in many configurations.
Answer by ikleyn(53250)  (Show Source):
You can put this solution on YOUR website! .
In the diagram, ABCD and AEFG are squares with side length 1.
Find the area of the green region.
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There is no diagram in this post, so we, the readers,
don't know what this problem is about.
Such kind of problems are considered in school Math as defective
and they are not a subject for consideration/discussions.
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