Q11
A regular octagon ABCDEFGH has sides of length 6 cm each. Then, the area, in sq. cm, of the square ACEG is
Answer: C
Asked: area of square ACEG inside a regular octagon of side 6.
Key idea: each interior angle of a regular octagon is 135°. AC is the side of the square, and it is the third side of triangle ABC with AB = BC = 6 and angle B = 135°.
Cosine rule: \(AC^2=6^2+6^2-2\cdot6\cdot6\cos135^\circ=72+72\cdot\dfrac{\sqrt2}{2}=72+36\sqrt2\).
Area of the square \(=AC^2=36(2+\sqrt2)\).
Check: the octagon's area is \(2(1+\sqrt2)\cdot36\approx173.8\); the square \(\approx122.9\) is smaller, as it should be.
Answer: \(36(2+\sqrt2)\)