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