Introduction

The relationship between an inscribed equilateral triangle and a square presents an interesting geometric problem with multiple solutions, each yielding different areas. This article will explore the various configurations of this problem and delve into the mathematical derivation of the areas for each case.

Case 1: Maximum Area of the Square

The most straightforward configuration is the scenario where the equilateral triangle is inscribed in a square such that its sides align perfectly with the square's sides. In this case, the area of the square is simply the square of its side length. Given the side of the equilateral triangle is 6 cm, the side of the square is 6 cm as well, yielding a square area of 36 cm^2.

Mathematical Derivation

Let's derive this using basic geometry: Given the side of an equilateral triangle is 6 cm, the maximum area of the square in which it can be inscribed is 6 x 6 36 cm^2. To verify this with an altitude, the altitude of an equilateral triangle with a side of 6 cm is given by 6 sin(60°) 3sqrt{3} cm ≈ 5.196 cm. This indicates that the height does not align with the side of the square, but the formula still yields the correct square area.

Case 2: Smallest Area of the Square

A more complex scenario involves finding the smallest possible square in which an equilateral triangle can be inscribed. One method is to orient the triangle such that its height aligns with the diagonal of the square.

Mathematical Derivation

1. **Calculate the height of the equilateral triangle:**
h 6 sin(60°) 3sqrt{3} cm ≈ 5.196 cm
The area of the square where this height aligns with the diagonal can be derived as follows:

Let the side length of the square be denoted by a. The height aligning with the diagonal means a^2 2h^2. Substitute h 3sqrt{3}, leading to a^2 2(3sqrt{3})^2 2 times 27 54. Hence, a sqrt{54} ≈ 7.35 cm

This results in an area of approximately 54 cm^2. However, further analysis shows that the smallest possible area occurs when the height of the triangle aligns with the diagonal as described.

Case 3: Specific Oritentation of the Triangle

Another interesting case is when the equilateral triangle is inscribed in a square but the vertex is in one corner of the square, touching all four sides. This requires a more intricate calculation involving trigonometric functions and the properties of the triangle and square.

Mathematical Derivation

1. **Calculate the side length of the square:**
Given the side of the triangle is 6 cm, and the angle at the corner is 15°, we use the cosine function to find the side length of the square.
side_{square} 6 cos(15°) ≈ 5.795 cm
This diagonal division requires more complex trigonometric calculations to determine the exact dimensions of the square.

Height and Area Calculation

1. **Calculate the height of the triangle:**
h 6 sin(60°) ≈ 5.196 cm
2. **Calculate the side of the square:**
BC sqrt{18} cos(15°) ≈ 5.792 cm
3. **Calculate the area of the square:**
Area_{square} (5.792)^2 ≈ 33.582 cm^2

Concluding Analysis

The problem of finding the area of a square in which an equilateral triangle is inscribed presents several interesting solutions. The minimum area is approximately 33.6 cm2, and the maximum area is 36 cm2. The exact value between these limits depends on the specific orientation and alignment of the triangle within the square.