find the difference between the area of regular hexagon plot each of whose side is 72 m and area of circular swimming tank inscribed in it

Let ABCDEF be the regular hexagon whose each side is 72 cm.

Sum of internal angles of the hexagon ABCDEF = (6 – 2) × 180° = 720°

Measure of each internal angle = 720° ÷ 6 = 120°

Since ABCDEF is regular hexagon, all diagonals will bisect the internal angles.

Therefore, in ΔOAB

∠OAB = ∠OBA = 60°

Also, ∠AOB = 180° – ∠OAB + ∠OBA = 180° – 120° = 60°

∴ ΔOAB is an equilateral triangle.

Let us draw OM⊥AB. Also, OM will bisect AB, ie, AM = MB = 36 cm.

In ΔOAB, by Pythagoras theorem

OA² = OM² + AM²

(72)² = (OM)² + (36)²

⇒ (OM)² = (72)² – (36)²

⇒ (OM)² = 5184 – 1296

⇒ (OM)² = 3888

⇒ OM = 62.35 (approximately)

Area of ABCDEF = 6 × ar (ΔOAB) (all six triangles inside the hexagon are identical)

 = 6 × 36 × 62.35

 = 13467.6 cm²

Radius of the circle will be the altitude from O to the side of the hexagon ie, equal to OM.

Radius of the circle (r) = 62.35 cm

Area of circle = πr² 

Difference between the areas of hexagon and circle = 13467.60 cm² – 12217.92 cm² = 1249.68 cm²

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