OR

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

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

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

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 ( $\pi r^2 = 962.5 \text{ cm}^2$ , with $r = 17.5 \text{ cm}$ ) equally among the 10 congruent sectors created by the 5 diameters (or equivalently using the sector formula with $\theta = 36^\circ$ ).

sector areaπr²10 equal sectors36 degreesbroochcircle area96.25 cm²

Marking Scheme

1½ mark: correctly finding radius $r = 17.5 \text{ cm}$ (or using it directly from diameter 35 cm).
2½ mark: total area of circle = $\pi r^2 = \left(\frac{22}{7}\right) \times 17.5^2 = 962.5 \text{ cm}^2$ .
3½ mark: correctly identifying 10 equal sectors (angle 36° each) and dividing appropriately.
4½ mark: correct final answer = $96.25 \text{ cm}^2$ per sector (with unit cm²).

Hint

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

Quick Oral Answer

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

Analysis & 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 $360^\circ/10 = 36^\circ$ .
Area of each sector = total circle area ÷ 10 (or directly by the sector formula).

Key points

$r = \frac{d}{2} = 17.5 \text{ cm}$ .
Total area = $\pi r^2 = \left(\frac{22}{7}\right) \times 17.5^2 = 962.5 \text{ cm}^2$ .
Area of each sector = $962.5 \div 10 = 96.25 \text{ cm}^2$ .

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.

Common Mistakes

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

Interesting Facts

A full circle always measures 360°, so dividing it into n equal sectors always gives each sector an angle of $360^\circ/n$ — here $360^\circ/10 = 36^\circ$ , 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 $(962.5 \text{ cm}^2)$ is a recurring 'nice' value in CBSE papers because $35^2 = 1225$ , and $\left(\frac{22}{7}\right) \times \frac{1225}{4} = 962.5$ , showing why multiples of 7 are favoured for diameter values.

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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 = $(\theta/360^\circ) \times \pi r^2$ with $\theta = 36^\circ$ gives the same answer, $96.25 \text{ cm}^2$ , and is an equally acceptable method for full marks.