Question 562 marksShort AnswerSection E

OR

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

Figure shows a circle of diameter 35 cm with 5 diameters drawn through the centre, dividing the circle into 10 equal sec
Fig. for Q56
Areas Related to Circles
Circles — Area of a Sector

Official answer

Area of each sector of the brooch = 96.25 cm². This is found by dividing the circle's total area (πr2=962.5 cm2\pi r^2 = 962.5 \text{ cm}^2, with r=17.5 cmr = 17.5 \text{ cm}) equally among the 10 congruent sectors created by the 5 diameters (or equivalently using the sector formula with θ=36∘\theta = 36^\circ).

Answer keywords:sector areaπr²10 equal sectors36 degreesbroochcircle area96.25 cm²

Marking scheme

  1. 1.½ mark: correctly finding radius r=17.5 cmr = 17.5 \text{ cm} (or using it directly from diameter 35 cm).
  2. 2.½ mark: total area of circle = πr2=(227)×17.52=962.5 cm2\pi r^2 = \left(\frac{22}{7}\right) \times 17.5^2 = 962.5 \text{ cm}^2.
  3. 3.½ mark: correctly identifying 10 equal sectors (angle 36° each) and dividing appropriately.
  4. 4.½ mark: correct final answer = 96.25 cm296.25 \text{ cm}^2 per sector (with unit cm²).

Hint

The 5 diameters create 10 equal sectors, each subtending an angle of 360∘/10=36∘360^\circ/10 = 36^\circ at the centre. Area of each sector = (110)×πr2=(36∘360∘)×πr2\left(\frac{1}{10}\right) \times \pi r^2 = \left(\frac{36^\circ}{360^\circ}\right) \times \pi r^2.

Quick oral answer

The 5 diameters split the circle into 10 equal sectors of 36° each; area of each sector = πr2/10=962.5/10=96.25 cm2\pi r^2/10 = 962.5/10 = 96.25 \text{ cm}^2.

Common mistakes

  1. 1.Dividing the total area by 5 (number of diameters) instead of 10 (number of sectors actually formed), giving double the correct area.
  2. 2.Using diameter instead of radius in the area formula πr2\pi r^2, i.e., mistakenly computing π×352\pi \times 35^2 instead of π×17.52\pi \times 17.5^2.
  3. 3.Forgetting to state the correct unit cm² for area, or leaving the answer as a fraction (77/... ) instead of the simplified decimal 96.25 cm296.25 \text{ cm}^2.

Analysis and explanation

Tests division of a circle's area into equal sectors formed by diameters drawn through the centre.


Concept

  • 5 diameters through the centre create 10 equal sectors, each of angle 360∘/10=36∘360^\circ/10 = 36^\circ.
  • Area of each sector = total circle area ÷ 10 (or directly by the sector formula).

Key points

  • r=d2=17.5 cmr = \frac{d}{2} = 17.5 \text{ cm}.
  • Total area = πr2=(227)×17.52=962.5 cm2\pi r^2 = \left(\frac{22}{7}\right) \times 17.5^2 = 962.5 \text{ cm}^2.
  • Area of each sector = 962.5÷10=96.25 cm2962.5 \div 10 = 96.25 \text{ cm}^2.

Common mistakes

  • Dividing by 5 instead of 10 (forgetting each diameter makes 2 sectors), giving double the correct area.
  • Squaring the diameter instead of the radius.

Real-world

  • Same idea used in dividing a circular pizza, clock face, or decorative motif into equal wedge-shaped parts.

Interesting facts

A full circle always measures 360°, so dividing it into n equal sectors always gives each sector an angle of 360∘/n360^\circ/n — here 360∘/10=36∘360^\circ/10 = 36^\circ, a fact widely used in pie-chart design in statistics as well.

CBSE frequently reuses the 'brooch' or 'pizza/pie' context for sector-area case studies because it visually and intuitively demonstrates equal angular division of a circle.

The area of a full circle with diameter 35 cm (962.5 cm2)(962.5 \text{ cm}^2) is a recurring 'nice' value in CBSE papers because 352=122535^2 = 1225, and (227)×12254=962.5\left(\frac{22}{7}\right) \times \frac{1225}{4} = 962.5, showing why multiples of 7 are favoured for diameter values.

Question details

Q56 of 56 questions

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

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Frequently asked questions

Why are there 10 sectors and not 5, if there are only 5 diameters?

Each diameter is a straight line passing through the centre and touching both sides of the circle, so it splits the circle into two sectors along its length; with 5 such diameters spaced apart, the circle gets divided into 10 equal sectors in total, each subtending 36° at the centre.

Can we use the direct sector formula instead of dividing total area by 10?

Yes — using Area of sector = (θ/360∘)×πr2(\theta/360^\circ) \times \pi r^2 with θ=36∘\theta = 36^\circ gives the same answer, 96.25 cm296.25 \text{ cm}^2, and is an equally acceptable method for full marks.