Question

Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is

• A.
  $3:4$
• B.
  $4:5$
• C.
  $2:3$
• D.
  $5:6$

Answer 1

The correct option is C

$2:3$


Let the side of the two larger equilateral triangle be 'a'

Then, side of regular hexagon will be
$=\left(\frac{a}{3}\right)$

Area of regular Hexagon
$=6\times \frac{\sqrt{3}}{4}{\left(\frac{a}{3}\right)}^2$

Area of triangle
$=\frac{\sqrt{3}}{4}(a{)}^2$

Required ratio
$=\frac{6\sqrt{3}}{4}\times \frac{a^2}{9}:\frac{\sqrt{3}}{4}{a}^2$

$=6:9=2:3$