A circle of radius is equal to the diagonal of the square whose perime

1.	A circle of radius is equal to the diagonal of the square whose perimeter is numerically √3 times the area of an equilateral triangle with circumradius 4√3 cm. Find double the area of the circle with that radius.1. 1458π cm22. 2916π cm23. 729π cm24. 2216π cm2
Answer» Correct Answer - Option 2 : 2916π cm2 Given: Radius of circle = Diagonal of square Perimeter of Square = √3 × area of equilateral triangle Circumradius of equilateral triangle = 4√3 cm Formulas Used: CircumRadius (R) = Side of equilateral triangle/√3 Area of Equilateral Triangle = (√3/4) × (Side)² Perimeter of Square = 4 × Side of square(a) Diagonal of Square = √2 × Side of Square Area of circle = π × (Radius)² Calculation: Side of equilateral triangle = √3 × 4√3 = 12 cm Area of Equilateral Triangle = (√3/4) × (12)² = 36√3 cm² Here it is given that perimeter of square is √3 times the Area of equilateral triangle Perimeter of Square = 4 × a = √3 × 36√3 = 108 cm Side of Square (a) = 108/4 = 27 cm Diagonal of Square = √2 × a = √2 × 27 = 27√2 cm Radius of Circle (r) = Diagonal of square ⇒ r = 27√2 cm Area of circle = π × (27√2)² = 1458π Double the area of circle = 2 × 1458π = 2916π ∴ Double the area of the circle is 2916π.

A circle of radius is equal to the diagonal of the square whose perimeter is numerically √3 times the area of an equilateral triangle with circumradius 4√3 cm. Find double the area of the circle with that radius.1. 1458π cm22. 2916π cm23. 729π cm24. 2216π cm2

Answer» Correct Answer - Option 2 : 2916π cm2

Given:

Radius of circle = Diagonal of square

Perimeter of Square = √3 × area of equilateral triangle

Circumradius of equilateral triangle = 4√3 cm

Formulas Used:

CircumRadius (R) = Side of equilateral triangle/√3

Area of Equilateral Triangle = (√3/4) × (Side)²

Perimeter of Square = 4 × Side of square(a)

Diagonal of Square = √2 × Side of Square

Area of circle = π × (Radius)²

Calculation:

Side of equilateral triangle = √3 × 4√3 = 12 cm

Area of Equilateral Triangle = (√3/4) × (12)² = 36√3 cm²

Here it is given that perimeter of square is √3 times the Area of equilateral triangle

Perimeter of Square = 4 × a = √3 × 36√3 = 108 cm

Side of Square (a) = 108/4 = 27 cm

Diagonal of Square = √2 × a = √2 × 27 = 27√2 cm

Radius of Circle (r) = Diagonal of square

⇒ r = 27√2 cm

Area of circle = π × (27√2)² = 1458π

Double the area of circle = 2 × 1458π = 2916π

∴ Double the area of the circle is 2916π.