A sheet of paper ‘PQRSTU’ is in the form of a regular hexagon having sides of length 2 cm as shown in the figure given below. What is the length of the side of the largest square sheet ‘ABCD’ that can be cut out from the given hexagonal sheet of paper?

Option 2 : 2√3(√3 - 1) cm

**Calculation:**

⇒ Let, AU = x cm

∴ AP + PB = 2(2 – x) cm

⇒ Consider the trapezium TUAD

⇒ ∠UAD = 180° - ∠AUT = 60°

⇒ Also, UA = TD

⇒ ∴ AD = UT + 2UAcos60° = 2 + x

⇒ In ∆APB:- AB = 2AP cos30° = 2(2 - x)√3/2

⇒ Now, AB = AD ⇒ 2 + x = (2 - x)√3

⇒ x = [2(√3 - 1)]/(√3 + 1)

⇒ AD = 2 + [2(√3 - 1)]/(√3 + 1) = 4√3 /(√3 + 1) cm

⇒ AD = 2√3(√3 - 1) cm

__Additional Information__

**Regular Hexagon**

⇒ It has 6 equal sides and 6 equal angles.

⇒ It has 6 vertices.

⇒ Sum of interior angles equals 720°.

⇒ Interior angle is 120° and exterior angle is 60°.

⇒ It is made up of six equilateral triangles.

⇒ 9 diagonals can be drawn inside a regular hexagon.

⇒ All the sides opposite to each other are parallel.

⇒ Area of Regular Polygon

A = 3√3/2 × a^{2}

Where a is the measurement of its sides or length of the side.

⇒ Perimeter of Hexagon

P = 6a