C
CraftExam
medium difficulty5 min

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.

In one line: Perimeter of sector = arc plus two spokes.

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1If r = 14 cm and θ = 90°, arc length = 22 cm using π = 22/7.
  2. 2Perimeter = 22 + 28 = 50 cm.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

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

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find the boundary length of a pie-slice shape or a sector-shaped garden piece.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    So its perimeter is not only the arc length.

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