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

What should be dimensions of the rectangles?

Given: ABHCDE is a regular hexagon made by using a rectangle and some pieces of rectangles cut along its diagonal.

We know that all the sides and angles of a regular hexagon are equal.

∴ AB = BH = HC = CD = DE = EA = 30 cm

Measure of each angle of a regular hexagon = 120°

⇒ ∠EDC = ∠DCH = ∠CHB = ∠HBA= ∠BAE = ∠AED = 120°

Measure of each angle of a rectangle = 90°

⇒ ∠ADC = ∠EFD = 90°

Now, ∠EDF = ∠EDC − ∠ADC = 120° − 90° = 30°

Using angle sum property in ΔEFD:

∠EFD + ∠FDE + ∠DEF = 180° 

⇒ 90° + 30° + ∠DEF = 180°

⇒ 120° + ∠DEF = 180°

⇒ ∠DEF = 180° − 120° = 60°

Thus, the angles of ΔEFD are 30°, 60° and 90°.

Therefore, the ratio of the lengths of the sides of ΔEFD is 1::2.

⇒ DF =

⇒ AF = DF = (As both are the sides of identical rectangles.)

AD = AF + FD =

EF =

Therefore, the dimensions of the larger rectangle are 30 cm and

The dimensions of the smaller rectangle are 15 cm and