Question

An equilateral triangle is cut from its three vertices to form a regular hexagon. What is the percentage of area wasted? (a) \(20 \%\) (b) \(50 \%\) (c) \(33.33 \%\) (d) \(66.66 \%\)

Short answer

Answer: (c) 33.33%

Step-by-step solution

Area of the initial equilateral triangle

Let's consider the side length of the equilateral triangle as 'a'. The area of an equilateral triangle can be found using the formula: Area = \(\frac{\sqrt{3}}{4} \cdot a^2\)

Area of a single removed triangle

After cutting off the vertices, we get a regular hexagon. In a hexagon, we can draw 6 equilateral triangles with the same side length as the original equilateral triangle. This means that the side length of each removed triangle is \(\frac{a}{3}\). Using the formula for the area of an equilateral triangle, we find the area of a single removed triangle: Area_removed_triangle = \(\frac{\sqrt{3}}{4} \cdot (\frac{a}{3})^2 = \frac{\sqrt{3}}{36} \cdot a^2\)

Calculate the total area of the removed triangles

Now, we need to find the total area of all three removed triangles. As each of the triangles has the same area, we can calculate the total area by multiplying the area of a single removed triangle by 3: Total_area_removed = \(3 \cdot \frac{\sqrt{3}}{36} \cdot a^2 = \frac{\sqrt{3}}{12} \cdot a^2\)

Calculate the area of the regular hexagon

The area of the regular hexagon can be found by subtracting the total area of the removed triangles from the area of the initial equilateral triangle: Area_hexagon = \(\frac{\sqrt{3}}{4} \cdot a^2 - \frac{\sqrt{3}}{12} \cdot a^2\)

Calculate the percentage of area wasted

Now, we need to find the percentage of area wasted. To do this, we divide the total area of the removed triangles by the area of the initial equilateral triangle, and then multiply the result by 100: Percentage_area_wasted = \(\frac{Total\_area\_removed}{Area\_of\_initial\_equilateral\_triangle} \cdot 100\) Percentage_area_wasted = \(\frac{\frac{\sqrt{3}}{12} \cdot a^2}{\frac{\sqrt{3}}{4} \cdot a^2} \cdot 100 = \frac{1}{3} \cdot 100 = 33.33 \%\)

Choose the correct answer

The percentage of area wasted is \(33.33\%\). Therefore, the correct answer is (c) \(33.33\%\).

Key concepts

Understanding Area Calculation

Calculating areas in geometry often involves using specific formulas based on the shapes we are dealing with. An equilateral triangle, having all sides of equal length, has its area calculated with the formula:

• Area of an equilateral triangle = \( \frac{\sqrt{3}}{4} \cdot a^2 \)

Here, 'a' represents the side of the triangle.
This formula is derived from the triangle’s symmetrical properties and is essential for solving various geometric problems.
In our problem, understanding this formula helps in calculating both the area of the initial equilateral triangle and the smaller triangles formed when a regular hexagon is carved out of the original shape.

Getting the Hang of Percentage Calculation

Percentage calculations are key when we want to compare parts of a whole. In geometry problems, this can mean finding out how much of a shape's original area is removed or altered.
To find a percentage, you use the formula:

• Percentage of a part = \( \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \)

In our scenario, the 'part' is the combined area of the removed triangles, and the 'whole' is the area of the original equilateral triangle.
This method allows us to understand what fraction of the triangle's area is wasted after forming a regular hexagon.

What Makes a Regular Hexagon Special?

A regular hexagon is a six-sided polygon where all sides and angles are equal.
When forming a hexagon from an equilateral triangle, you essentially are removing smaller triangles from its vertices.
Due to the symmetry of a regular hexagon, it can be neatly divided into six smaller equilateral triangles. This characteristic is utilized to simplify the process of calculating areas when shapes interact or change.
Recognizing these properties helps in understanding why subtracting certain triangular areas results in a hexagon, and how to calculate remaining areas efficiently.

Cracking the Code of Geometry Problem Solving

Approaching geometry problems requires a clear understanding of spatial relationships and properties of shapes. Here’s a structure to help in solving complex geometry problems:

• Identify all shapes involved. Understand their properties and theorems associated with them.
• Break down the problem into manageable steps, such as calculating individual areas before combining them.
• Use known formulas to determine key values. Be precise in algebraic manipulation to avoid errors.
• Reassess steps to ensure each has been calculated correctly, particularly when involving conversions like percentages.

By following these steps, you can efficiently solve for areas, percentages, and transformations within geometric figures, making even complicated exercises much clearer.