An equilateral triangle is cut from its three vertices to form a regular hexagon . What is the percentage of area wasted ?

Options

A. A)20%
B. B)50%
C. C)33.33%
D. D)66.66%

Answer

To solve the problem of finding the percentage of area wasted when an equilateral triangle is cut from its three vertices to form a regular hexagon, we can follow these steps:

### Step 1: Understand the Geometry
We start with an equilateral triangle and cut off its three vertices to form a regular hexagon. Let's denote the side length of the equilateral triangle as \( A \).

### Step 2: Calculate the Area of the Equilateral Triangle
The formula for the area \( A_T \) of an equilateral triangle with side length \( A \) is given by:
\[ A_T = \frac{\sqrt{3}}{4} A^2 \]
For our example, let’s assume \( A = 3 \) cm. Thus, the area of the triangle becomes:
\[ A_T = \frac{\sqrt{3}}{4} \times 3^2 = \frac{\sqrt{3}}{4} \times 9 = \frac{9\sqrt{3}}{4} \text{ cm}^2 \]

### Step 3: Determine the Side Length of the Hexagon
When we cut the vertices of the triangle, we create a regular hexagon. Each side of the hexagon corresponds to the segments created by cutting the triangle. Since the triangle is equilateral, each side of the hexagon will be \( 1 \) cm (since the original side of the triangle is divided into three equal parts).

### Step 4: Calculate the Area of the Regular Hexagon
The formula for the area \( A_H \) of a regular hexagon with side length \( s \) is given by:
\[ A_H = \frac{3\sqrt{3}}{2} s^2 \]
Substituting \( s = 1 \) cm:
\[ A_H = \frac{3\sqrt{3}}{2} \times 1^2 = \frac{3\sqrt{3}}{2} \text{ cm}^2 \]

### Step 5: Calculate the Area Wasted
The area wasted \( A_W \) is the difference between the area of the triangle and the area of the hexagon:
\[ A_W = A_T - A_H \]
Substituting the areas calculated:
\[ A_W = \frac{9\sqrt{3}}{4} - \frac{3\sqrt{3}}{2} \]
To subtract these, we need a common denominator:
\[ A_H = \frac{3\sqrt{3}}{2} = \frac{6\sqrt{3}}{4} \]
Thus,
\[ A_W = \frac{9\sqrt{3}}{4} - \frac{6\sqrt{3}}{4} = \frac{3\sqrt{3}}{4} \text{ cm}^2 \]

### Step 6: Calculate the Percentage of Area Wasted
To find the percentage of area wasted, we use the formula:
\[ \text{Percentage Wasted} = \left( \frac{A_W}{A_T} \right) \times 100 \]
Substituting the areas:
\[ \text{Percentage Wasted} = \left( \frac{\frac{3\sqrt{3}}{4}}{\frac{9\sqrt{3}}{4}} \right) \times 100 = \left( \frac{3}{9} \right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer
The percentage of area wasted is **33.33%**.

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