Shortcut Trick Ratio = 2πr : [2r + (θ/360) × 2πr]

Substitute θ = 30°: Ratio = 2π : [2 + (30/360) × 2π]

Ratio = 2π : [2 + π/6] ⇒ Ratio = 12π : (π + 12)

Using π = 3.14: Ratio = (12 × 3.14) : (3.14 + 12)

Ratio = 37.68 : 15.14 = 3768 : 1514 = 1884 : 757

∴ The correct answer is 1884 : 757.

Alternate Method

Given:

Central angle of sector (θ) = 30°

Value of π = 3.14

Formula Used:

Circumference of Circle (C) = 2 × π × r

Perimeter of Sector (P) = (Length of Arc) + (2 × Radius)

Length of Arc = (θ/360) × 2 × π × r

Calculations:

⇒ Circumference (C) = 2πr

⇒ Arc Length = (30/360) × 2πr = (1/12) × 2πr = πr/6

⇒ Perimeter of Sector (P) = (πr/6) + 2r

⇒ P = r[(π/6) + 2] = r[(π + 12)/6]

⇒ Ratio = C : P = 2πr : r[(π + 12)/6]

⇒ Ratio = 2π : (π + 12)/6

⇒ Ratio = 12π : (π + 12)

⇒ Ratio = (12 × 3.14) : (3.14 + 12)

⇒ Ratio = 37.68 : 15.14

⇒ Ratio = 3768 : 1514 = 1884 : 757

∴ The correct answer is 1884 : 757.

Additional Information

Area of Sector

The area of a sector with central angle θ is given by Area = (θ/360) × πr².

Relation between Area and Arc Length

The area of a sector can also be expressed as Area = (1/2) × l × r, where l is the arc length and r is the radius.

Perimeter of a Semicircle

The perimeter of a semicircle is πr + 2r, which is a specific case of a sector where θ = 180°.