A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm , of the square ACEG
is

Question

A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm , of the square ACEG
 is

UPSC Quizroom · Accepted Answer

Step 1: Interior angle of a regular octagon

Formula: Interior angle = ((n−2)/n) × 180°
n = 8 → ((8−2)/8) × 180° = (6/8) × 180° = 135°

Step 2: Area of square ACEG

Let side length = 6 cm → a = 6

In ΔBAC (triangle formed by consecutive vertices), use cosine rule to find diagonal:

c² = a² + b² − 2ab cos C
c² = 6² + 6² − 2·6·6·cos135°
c² = 36 + 36 − 72 × (−√2/2)
c² = 72 + 36√2
c = √[72 + 36√2]

Area of square ACEG = (diagonal)² / 2 = c² / 2 = 36(2 + √2)

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