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We are tasked with finding the area of region \( R \), which lies inside the unit square \( ABCD \) but outside equilateral triangle \( ABE \), and whose distance from side \( AD \) is between \( \frac{1}{3} \) and \( \frac{2}{3} \).

### Step 1: Understanding the Geometry
- The square \( ABCD \) has side length 1, meaning the area of the square is 1.
- Triangle \( ABE \) is equilateral, with one vertex at \( A \) (a corner of the square) and another vertex \( E \) inside the square. However, we do not need the precise location of vertex \( E \) to compute the required area of \( R \).
  
### Step 2: The Strips Defined by the Distance from Side \( AD \)
- We are focusing on the region that lies within the unit square but outside triangle \( ABE \), and whose distance from side \( AD \) is between \( \frac{1}{3} \) and \( \frac{2}{3} \).
- The strip formed by points whose distance from side \( AD \) is between \( \frac{1}{3} \) and \( \frac{2}{3} \) is a horizontal band across the square with width \( \frac{1}{3} \), extending from \( y = \frac{1}{3} \) to \( y = \frac{2}{3} \).

### Step 3: Subtracting the Area of the Triangle Within the Strip
- The area of the horizontal strip between \( \frac{1}{3} \) and \( \frac{2}{3} \) is \( 1 \times \left( \frac{2}{3} - \frac{1}{3} \right) = \frac{1}{3} \).
- However, we must exclude the portion of triangle \( ABE \) that lies within this strip.
  
### Step 4: Conclusion
Since triangle \( ABE \) lies in the bottom left corner and vertex \( E \) is inside the square, the portion of the triangle within the strip is relatively small. Given that the area of the entire triangle \( ABE \) is small compared to the square, the area of \( R \) (the region between the two distances from \( AD \) and outside the triangle) will be dominated by the area of the strip itself.

Thus, the approximate area of region \( R \) is:

\[
\boxed{\frac{1}{3}}
\]

Would you like further details on any step or another related explanation?

Here are 5 questions you might explore next:
1. How do you compute the area of an equilateral triangle inside a unit square?
2. What if the triangle \( ABE \) were not equilateral—how would the solution change?
3. How would the problem change if the distances from \( AD \) were different, say \( \frac{1}{4} \) and \( \frac{3}{4} \)?
4. How do we calculate the area of a region inside a square but outside multiple shapes?
5. What is the formula for the area of a general region in a unit square defined by strips?

**Tip:** Understanding how strips within geometric figures work helps in calculating areas of irregular shapes!

Vertex E of equilateral triangle ABE is in the interior of unit square ABCD. Let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/3 and 2/3. What is the area of R?

Learn how to find the area of a region inside a unit square and outside an equilateral triangle, defined by specific distances from a side. Explore the geometry of squares, triangles, and strips in this detailed problem solution. Suitable for advanced high school students.

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