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An equilateral triangle ABC of side 24 cm is cut out to form a regular hexagon PQRSTU as shown in the figure. Find the area of the hexagon in cm2.

An equilateral triangle ABC of side 24 cm is cut out to form a regular hexagon PQRSTU as shown in the figure. Find the area of the hexagon in cm2. Correct Answer 96√3

GIVEN:

Side of equilateral triangle = 24 cm

FORMULA USED:

Area of the hexagon = 6 ×

CALCULATION:

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Each interior angle of the regular hexagon is 120°

⇒ Each external angle = 180° - 120° = 60°,

The small triangles that are cut off (triangle UBT, triangle RSC and triangle APQ) are all equilateral triangle since all the angles are 60°.

Let the side of the hexagon be x cm

So,

⇒ AP = AQ = side of the hexagon = x

Similarly,

⇒ BU = UT = side of the hexagon = x

⇒ CR = RS = side of the hexagon = x

Now,

⇒ AB = AP + x + UB = 24

⇒ 3x = 24

⇒ x = 8 cm

Hence,

Area of the hexagon = 6 × (√3/4x2)

= 6 × (√3/4) × 82

= 96√3 cm2

∴ The area of the hexagon is 96√3 cm2

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