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Length of an Arc and Sector Perimeter

Arc length is the length of a part of a circle's circumference, and sector perimeter is the arc length plus two radii.

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Student-friendly explanation

A sector is like a slice of a circle. Its curved part is only a fraction of the full circumference, decided by the central angle.

How to write this in exams

  1. 1

    Start with the exact idea

    Arc length is the length of a part of a circle's circumference, and sector perimeter is the arc length plus two radii.

  2. 2

    Then show how to use it

    Find the fraction θ/360, multiply by full circumference, then add two radii if perimeter is asked.

  3. 3

    Add one concrete example

    For radius 14 cm and angle 90°, arc length = 90/360 × 2 × 22/7 × 14 = 22 cm; sector perimeter = 22 + 14 + 14 = 50 cm.

  4. 4

    Avoid this incomplete answer

    22 cm for the perimeter of a 60°, 21 cm sector is wrong because it gives only the arc length.

Definition

Arc length is the length of a part of a circle's circumference, and sector perimeter is the arc length plus two radii.

Example

For radius 14 cm and angle 90°, arc length = 90/360 × 2 × 22/7 × 14 = 22 cm; sector perimeter = 22 + 14 + 14 = 50 cm.

Rule to remember

Arc length = θ/360 × 2πr. Sector perimeter = arc length + 2r, where θ is the central angle in degrees.

Memory hook

Sector boundary has one curve and two radii.

Examples and method

Worked example

For θ = 60° and r = 21 cm, arc length = 60/360 × 2 × 22/7 × 21 = 22 cm. Sector perimeter = 22 + 42 = 64 cm.

Method to apply

Find the fraction θ/360, multiply by full circumference, then add two radii if perimeter is asked.

Diagram support

Draw a circle sector with central angle θ, radius r on both straight sides, and the curved arc labelled.

How CBSE asks it

Exams may show a sector diagram and ask for arc length, perimeter, or missing radius or angle.

Avoid common mistakes

Common confusion

Students often give only arc length when the question asks for the perimeter of a sector.

Common wrong answer

22 cm for the perimeter of a 60°, 21 cm sector is wrong because it gives only the arc length.

Exam tip

For sector perimeter, remember to add the two straight radii after finding the arc length.

Quick check

Why is the perimeter of a sector not equal to only its arc length?

The perimeter of a sector includes the curved arc and the two radii that form the sides of the sector.

Answer writing and exam use

1-mark answer

Arc length is the length of a part of a circle's circumference, and sector perimeter is the arc length plus two radii.

2-mark answer

Arc length is the length of a part of a circle's circumference, and sector perimeter is the arc length plus two radii. Arc length = θ/360 × 2πr. Sector perimeter = arc length + 2r, where θ is the central angle in degrees. For radius 14 cm and angle 90°, arc length = 90/360 × 2 × 22/7 × 14 = 22 cm; sector perimeter = 22 + 14 + 14 = 50 cm.

3-mark answer

A sector is like a slice of a circle. Its curved part is only a fraction of the full circumference, decided by the central angle. Arc length = θ/360 × 2πr. Sector perimeter = arc length + 2r, where θ is the central angle in degrees. For θ = 60° and r = 21 cm, arc length = 60/360 × 2 × 22/7 × 21 = 22 cm. Sector perimeter = 22 + 42 = 64 cm. Exams may show a sector diagram and ask for arc length, perimeter, or missing radius or angle. 22 cm for the perimeter of a 60°, 21 cm sector is wrong because it gives only the arc length.
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