Question

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

• A. $25$%
• B. $66.66$%
• C. $33.33$%
• D. $50$%

Answer 1

Answer: (A)

$25$%

Say we have an equilateral triangle of side $3a$. To create a regular hexagon of length ${}^{\prime }{a}^{\prime }$, one needs to cut an equilateral triangle from each vertex.

So the shaded portion is cut off.

Each of the shaded region is again an equilateral triangle of area $\frac{\sqrt{3}}{4}{a}^2$

And total area of the triangle is $\frac{\sqrt{3}}{4}{\left(3a\right)}^2=9\frac{\sqrt{3}}{4}{a}^2$

$\therefore$ Area wasted is $3\times \left(\frac{\sqrt{3}}{4}{a}^2\right)=3\frac{\sqrt{3}}{4}{a}^2$

% of area wasted $=\frac{3\frac{\sqrt{3}}{4}{a}^2}{9\frac{\sqrt{3}}{4}{a}^2}\times 100=\frac{1}{3}\times 100=33.33$%