Question

Three equilateral triangles are cut from an equilateral triangle of area 36 $c{m}^2$ such that a regular hexagon is left. Find the area of hexagon (in $c{m}^2$ )

• A. $24$
• B. $28$
• C. $30$
• D. $32$

Answer 1

Answer: (A)

$24$

Given that:

Area of $△ABC=36c{m}^2$

To find:

Area of the hexagon$=?$

Solution:

Let ${}^{\prime }{a}^{\prime }$ be length of the side equilateral $△ABC$

Then, length of the side of the smaller equilateral triangle $=\frac{a}{3}$

$ar\left(△ABC\right)=36c{m}^2$

or, $\frac{\sqrt{3}}{4}{a}^2=36c{m}^2$

or, $a^2=\frac{36\times 4}{\sqrt{3}}c{m}^2$

Now,

Area of the hexagon= Area of $△ABC-3\times$ Area of smaller triangle

$=36c{m}^2-3\times \frac{\sqrt{3}}{4}\times {\left(\frac{a}{3}\right)}^2$

$=36c{m}^2-\frac{\sqrt{3}}{4}\times \frac{36\times 4}{3\times \sqrt{3}}c{m}^2$

$=36c{m}^2-12c{m}^2$

$=24c{m}^2$

So, Area of the hexagon $=24c{m}^2$

Hence, A is the correct option.