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Perimeter of a sector

The perimeter of a sector is the total boundary length of the sector, including the two radii and the curved arc.

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Student-friendly explanation

A sector has three boundary parts: one curved arc and two straight radii. So its perimeter is not only the arc length. Students must add the arc length and the two radii together to get the full boundary.

How to write this in exams

  1. 1

    Start with the exact idea

    The perimeter of a sector is the total boundary length of the sector, including the two radii and the curved arc.

  2. 2

    Then show how to use it

    1. Find the arc length first if needed. 2. Identify the radius. 3. Add 2r for the two straight sides. 4. Combine carefully. 5. Write the answer as a length.

  3. 3

    Add one concrete example

    For a sector with radius 6 cm and arc length 5π cm, perimeter = 5π + 12 cm.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 22 cm for the example above, where the student forgets to include both radii.

Definition

The perimeter of a sector is the total boundary length of the sector, including the two radii and the curved arc.

Example

For a sector with radius 6 cm and arc length 5π cm, perimeter = 5π + 12 cm.

Rule to remember

Perimeter of sector = arc length + 2r = (θ/360) × 2πr + 2r.

Memory hook

Perimeter of sector = arc plus two spokes.

Examples and method

Worked example

If r = 14 cm and θ = 90°, arc length = 22 cm using π = 22/7. Perimeter = 22 + 28 = 50 cm.

Method to apply

1. Find the arc length first if needed. 2. Identify the radius. 3. Add 2r for the two straight sides. 4. Combine carefully. 5. Write the answer as a length.

Diagram support

Draw the sector and label the two radii as straight sides, then highlight the curved arc as the third boundary part.

How CBSE asks it

Find the boundary length of a pie-slice shape or a sector-shaped garden piece.

Avoid common mistakes

Common confusion

Students often write only the arc length and forget to add the two radii.

Common wrong answer

A common wrong answer is 22 cm for the example above, where the student forgets to include both radii.

Exam tip

Whenever the word perimeter appears, think total boundary, not just the curved part.

Quick check

A sector has radius 9 cm. If its arc length is 10 cm, what else must be added to get the perimeter?

You must add the two radii, so add 9 cm + 9 cm. The perimeter is the arc length plus these two straight sides.

Answer writing and exam use

1-mark answer

The perimeter of a sector is the total boundary length of the sector, including the two radii and the curved arc.

2-mark answer

The perimeter of a sector is the total boundary length of the sector, including the two radii and the curved arc. Perimeter of sector = arc length + 2r = (θ/360) × 2πr + 2r. For a sector with radius 6 cm and arc length 5π cm, perimeter = 5π + 12 cm.

3-mark answer

A sector has three boundary parts: one curved arc and two straight radii. So its perimeter is not only the arc length. Students must add the arc length and the two radii together to get the full boundary. Perimeter of sector = arc length + 2r = (θ/360) × 2πr + 2r. If r = 14 cm and θ = 90°, arc length = 22 cm using π = 22/7. Perimeter = 22 + 28 = 50 cm. Find the boundary length of a pie-slice shape or a sector-shaped garden piece. A common wrong answer is 22 cm for the example above, where the student forgets to include both radii.
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