Question 532 marksSection E

  1. (iii) (b) What is the area of each sector of the brooch ?

Areas Related to Circles
Area of a sector

Official answer

Area of each sector=(total circle area)÷10=πr210=(227×17.5×17.5)10=962.510=96.25 cm2\text{Area of each sector} = (\text{total circle area}) \div 10 = \frac{\pi r^2}{10} = \frac{\left(\frac{22}{7} \times 17.5 \times 17.5\right)}{10} = \frac{962.5}{10} = 96.25 \text{ cm}^2. (Equivalently, using θ=36°:(36360)×πr2=96.25 cm2\theta = 36°: \left(\frac{36}{360}\right) \times \pi r^2 = 96.25 \text{ cm}^2.)

Answer keywords:area of sector(θ/360)πr²36 degrees10 equal sectors96.25 cm²πr² = 962.5brooch

Marking scheme

  1. 1.1 mark: correct area of the full circle πr2=962.5 cm2\pi r^2 = 962.5 \text{ cm}^2, or correct sector angle θ=36°\theta = 36° with the sector formula (θ/360)πr2(\theta/360)\pi r^2.
  2. 2.1 mark: correct area of each sector=96.25 cm2\text{area of each sector} = 96.25 \text{ cm}^2 with unit.
  3. 3.Accept dividing total area by 10, or using θ=36°\theta = 36° — both must give 96.25 cm296.25 \text{ cm}^2.
  4. 4.Only 1 mark if the full circle area is correct but not divided by 10 (i.e., 962.5 left as the final answer).

Hint

10 equal sectors means each sector is one-tenth of the whole circle's area (or use θ=36°\theta = 36° in (θ/360)πr2(\theta/360)\pi r^2).

Quick oral answer

Five diameters make ten equal sectors, so each sector is one-tenth of the circle's area, that is 962.5 divided by 10, which equals 96.25 square centimetres.

Common mistakes

  1. 1.Giving the whole circle area (962.5 cm2962.5 \text{ cm}^2) as the sector area — forgetting to divide by 10.
  2. 2.Using θ=360/5=72°\theta = 360/5 = 72° (5 sectors) instead of θ=360/10=36°\theta = 360/10 = 36°, since 5 diameters make 10 sectors, not 5.
  3. 3.Squaring the diameter or the wrong radius — must use r=17.5 cmr = 17.5 \text{ cm}, and (17.5)2=306.25(17.5)^2 = 306.25.

Analysis and explanation

This 2-mark part tests the sector-area formula, but the real trick is counting sectors correctly.


Concept

  • 5 diameters passing through the centre cut the circle into 10 (not 5) equal sectors, each with central angle 360°/10=36°360°/10 = 36°.
  • Area of each sector=(θ360)×πr2\text{Area of each sector} = \left(\frac{\theta}{360}\right) \times \pi r^2, or equivalently the whole circle's area divided by 10.

Common mistakes

  • The '5 vs 10' confusion — using 72°72° instead of 36°36°, which doubles the true sector area.
  • Forgetting to divide by the number of sectors and reporting the full circle area (962.5 cm2962.5 \text{ cm}^2) as the answer.
  • Using diameter instead of radius inside πr2\pi r^2, which quadruples the area.

Real-world

  • Knowing one sector's area lets a jeweller estimate the silver-sheet or enamel needed for a single wedge; multiplying back by 10 gives a built-in consistency check (10×96.25=962.5 cm210 \times 96.25 = 962.5 \text{ cm}^2).

Interesting facts

5 diameters cut the circle into 10 sectors, each subtending 36°36° — the same angle between adjacent points of a regular 10-pointed star.

Because the sectors are equal, area per sector=total area÷10\text{area per sector} = \text{total area} \div 10 is far quicker than the (θ/360)πr2(\theta/360)\pi r^2 formula, though both give 96.25 cm296.25 \text{ cm}^2.

The 10 sector areas add back to 962.5 cm2962.5 \text{ cm}^2, a good self-check that the parts reconstitute the whole circle.

Question details

Q53 of 53 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Basic)
Year
2026
Section
Section E
Marks
2
Chapter
Areas Related to Circles
Topic
Area of a sector
Bloom's level
apply
Syllabus topic
Unit 5: Mensuration — Areas Related to Circles: Area of a Sector of a Circle
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What is the central angle of each sector?

5 diameters create 10 equal sectors, so each central angle is 360°/10=36°360°/10 = 36°. A common slip is to write 72°72° by dividing by 5, but 5 diameters give 10 divisions, not 5.

What is the fastest way to get the sector area?

Since all 10 sectors are equal, each is one-tenth of the whole circle: πr2/10=962.5/10=96.25 cm2\pi r^2/10 = 962.5/10 = 96.25 \text{ cm}^2. This avoids the (θ/360)πr2(\theta/360)\pi r^2 formula, though that gives the same 96.25 cm².