Question 9 of 42intermediate🔧 ApplyNumerical3 marks

Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius 14 cm and sector angle 75°.

Correct Answer

46 1/3 cm (or 139/3 cm)

Exercise: Exercise Set 6.1 | Q: 4 | (Page 12)
For More Understanding

Explanation

The textbook defines the perimeter of a sector as the sum of the curved portion (the arc) and the two straight portions (the two radii). Using the rotational symmetry of the circle as discussed in the text, the length of the arc is a proportional fraction (θ/360) of the total circumference (2πr). Adding the lengths of the two radii (2r) to this arc length gives the total perimeter of the sector.

Solution Steps

  1. Step 1: Identify the components of the perimeter. The perimeter of a sector is the sum of the curved portion (arc length) and the two straight portions (2 radii). So, Perimeter = Arc length + 2r.

  2. Step 2: Calculate the arc length. Using the rotational symmetry of the circle, Arc length = (θ/360) × 2πr. Substituting θ = 75°, r = 14 cm, and π = 22/7, we get Arc length = (75/360) × 2 × (22/7) × 14 = (5/24) × 88 = 55/3 cm.

  3. Step 3: Calculate the total perimeter. Perimeter = (55/3) + 2(14) = (55/3) + 28 = (55 + 84)/3 = 139/3 = 46 1/3 cm.

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