https://file.aiquickdraw.com/mathbot/file/1729844263268z5qxjn5v.jpg

The problem states:

1. A regular hexagon is inscribed in a circle of radius 6 centered at the origin.
2. The hexagon is reflected about the line \( y = x \) to form a second hexagon.
3. The task is to find the area of the overlap between the two hexagons.

### Step-by-Step Solution

1. **Hexagon Description**:
   A regular hexagon inscribed in a circle of radius \( 6 \) has all its vertices equidistant from the origin. The length of each side of the hexagon can be calculated using the formula:
   \[
   \text{Side length} = r \cdot \sin\left(\frac{\pi}{3}\right) = 6 \cdot \sin\left(\frac{\pi}{3}\right) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3}
   \]

   The area of a regular hexagon with a given radius \( r \) is:
   \[
   \text{Area of hexagon} = \frac{3\sqrt{3}}{2} r^2
   \]
   Substituting \( r = 6 \):
   \[
   \text{Area of each hexagon} = \frac{3\sqrt{3}}{2} \cdot 6^2 = \frac{3\sqrt{3}}{2} \cdot 36 = 54\sqrt{3}
   \]

2. **Reflection about \( y = x \)**:
   When a hexagon is reflected about the line \( y = x \), its vertices will be mirrored accordingly. The vertices that are symmetric with respect to \( y = x \) will have their coordinates swapped. For instance, if one vertex is at \( (a, b) \), its reflected vertex would be at \( (b, a) \).

3. **Determining Overlap**:
   The reflected hexagon and the original hexagon will share a common area, which needs to be found. Both hexagons are of equal size, and the centers of both are located symmetrically about the line \( y = x \).

4. **Symmetry Consideration**:
   The problem has symmetrical properties because of the reflection about \( y = x \). The resulting intersection of these two regular hexagons will form a smaller symmetric shape. For regular hexagons inscribed in a circle, the overlap will consist of six identical segments shared by both hexagons.

5. **Area Calculation**:
   The formula to find the area of overlap between two reflected regular hexagons in a case like this is:
   \[
   \text{Area of Overlap} = 2 \times (\text{Area of a Regular Triangle with side length } 6)
   \]
   The area of an equilateral triangle with side length \( s \) is:
   \[
   \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} s^2
   \]
   Using \( s = 6 \):
   \[
   \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}
   \]

   Since there are two such triangles contributing to the overlap:
   \[
   \text{Total Overlap Area} = 2 \times 9\sqrt{3} = 18\sqrt{3}
   \]

### Final Answer:
The area of the overlap between the two hexagons is \( \boxed{18\sqrt{3}} \).

---

Would you like more details or have any other questions about this solution?

Here are some related questions to consider:

1. How do you find the side length of a regular hexagon inscribed in a circle with a given radius?
2. What transformations occur when a figure is reflected about a line?
3. How does reflecting about \( y = x \) affect the coordinates of points?
4. What other shapes might result from intersecting regular polygons?
5. How can symmetry help in simplifying the calculation of overlapping areas?

**Tip**: Always use symmetry properties to reduce complex overlap calculations—reflections and rotations often preserve certain dimensions and angles.

A regular hexagon is inscribed in a circle centered at the origin with radius 6. The hexagon is then reflected about the line y = x to form a second hexagon. What is the area of the overlap between the two hexagons?

Find the area of the overlap between two regular hexagons, one inscribed in a circle with radius 6 and the other reflected about the line y = x. Learn about symmetry, reflection, and how to calculate the area of polygons with detailed explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation