Two rectangles are cut along the diagonal and the triangles got are to be joined to another rectangle to make a regular hexagon as shown below:

If the sides of the hexagon are 30 centimetres, what would be the length and breadth of the rectangles?

Let each polygon be assigned with name.

In regular hexagon AB₁F₁DF₂B₂A

AB₁ = B₁F₁ = F₁D = DF₂ = F₂B₂ = B₂A = 30 cm

∠B₂AB₁ = ∠AB₁F₁ = ∠B₁F₁D = ∠F₁DF₂ = ∠DF₂B₂ = ∠F₂B₂A = 120°

Δ AB₁G and Δ AB₂G are the two equal halves of one rectangle.
and Δ F₁DH and Δ F₂DH are two equal halves of another rectangle.

⇒ ∠B₁AG = ∠B₂AG

Now, ∠B₁AB₂ = ∠B₁AG + ∠B₂AG = 120°
⇒ ∠B₁AG = ∠B₂AG = 60°

In Δ AGB_1,

∠AGB₁ + ∠AB₁G + ∠GAB₁ = 180°
⇒ 90° + 60° + ∠GAB₁ = 180°
⇒ ∠GAB₁ = 30°

We know that sides of any triangle of angles 30°, 60° and 90°
are in the ratio 1:√3:2.

⇒ AG : B₁G: AB₁ = 1: √3: 2
⇒ AG: B₁G: 30 = 1: √3: 2
⇒ AG = 15 cm and B₁G = 15√3 cm

Similarly, In Δ AB₁B₂
⇒ B₁B₂ = B₁G + B₂G = 2(B₁G) = 2(15√3) cm = 30√3 cm

∴ For small rectangles :
Length = B₂G = B₁G = 15√3 cm
Breadth = AG = 15 cm

∴ For rectangle B₁B₂ H₂H₁ :
Length = B₁B₂ = 30√3 cm
Breadth = B₁F₁ = 30 cm

Length and Breadth of rectangles in cm are
(15√3 ,15),(15√3, 15) and ( 30√3, 30).