How to Find the Area of the Square Inside a Regular Octagon

Find the area of the square inside a regular hexagon, when the side of the regular octagon is 2 cm

From figure four squares are are inscribed inside a regular octagon, one side of the squares are equal to the side of the octagon, then find the area of the square inside a regular hexagon, when the side of the regular octagon is 2 cm
Solution

From the figure, Apply Pythagorean theorem in triangle AUH
Where AU = HU {because of symmetry}
Now, AH² = AU² + HU²
⇒ 2² = 2 × AU²
⇒ 2 = AU²
So, AU = HU = √2 cm
JT = 2 – AU = 2 – √2 cm
SI = JT = 2 – √2 cm {because of symmetry}
TS = JI – (JT + SI)
⇒ TS = 2 – (2 – √2 + 2 – √2)
⇒ TS = 2 – (4 – 2√2)
Thus, TS = 2√2 – 2 cm
Area of the blue square = TS² = (2√2 – 2)²
Thus, Area of the blue square = 12√2 – 8 cm²