Shortcut Trick
When the vertices of an equilateral triangle are cut to form a regular hexagon, the side length is exactly trisected.
Side of the regular hexagon = Side of triangle ÷ 3 = a / 3.
Area of a regular hexagon = 6 × Area of small equilateral triangle = 6 × (√3 ÷ 4) × (side)2.
Area = 6 × (√3 ÷ 4) × (a / 3)2 = (3√3 ÷ 2) × (a2 ÷ 9).
Area = √3a2 ÷ 6 = a2 / (2√3).
∴ The correct answer is a2 / (2√3) square units.
Alternate Method
Given:
An equilateral triangle with side length a.
The vertices are cut out to make a regular hexagon.
Formula Used:
Area of an equilateral triangle = (√3 ÷ 4) × a2
Area of a regular hexagon = 6 × (√3 ÷ 4) × x2 (where x is the side length)
Calculations:
⇒ Let the side length of the regular hexagon be x.
⇒ When cutting the vertices of an equilateral triangle to form a regular hexagon, the cut-out portions are smaller equilateral triangles of side length x.
⇒ The original side 'a' consists of three linear segments: one side of a cut-off triangle, one side of the hexagon, and one side of another cut-off triangle.
⇒ x + x + x = a
⇒ 3x = a
⇒ x = a / 3
⇒ The area of the regular hexagon is the combined area of 6 small equilateral triangles of side x.
⇒ Area = 6 × [(√3 ÷ 4) × (a / 3)2]
⇒ Area = 6 × (√3 ÷ 4) × (a2 ÷ 9)
⇒ Area = (6√3 × a2) ÷ 36
⇒ Area = (√3 × a2) ÷ 6
⇒ Multiplying numerator and denominator by √3 to simplify the fraction:
⇒ Area = (√3 × √3 × a2) ÷ (6 × √3) = 3a2 ÷ 6√3
⇒ Area = a2 / (2√3)
∴ The correct answer is a2 / (2√3) square units.
Additional Information
Area of an Equilateral Triangle
The area of an equilateral triangle with side 'a' is given by Area = (√3 ÷ 4) × a2, and its height is h = (√3 ÷ 2) × a.
Regular Hexagon Area
A regular hexagon with side 'x' is structurally made of 6 identical equilateral triangles. Its total area is 6 × (√3 ÷ 4) × x2 = (3√3 ÷ 2) × x2.
Inradius & Circumradius of Equilateral Triangle
For an equilateral triangle of side 'a', the inradius is r = a ÷ (2√3) and the circumradius is R = a ÷ √3. Note that the circumradius is always exactly double the inradius (R = 2r).
Max Area Hexagon in a Triangle
The largest regular hexagon that can be inscribed in any equilateral triangle is formed precisely by cutting off its corners, and its side length is always strictly one-third of the parent triangle's side.