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

Areas Related to Circles

Area of a sector

Official Answer

$\text{Area of each sector} = (\text{total circle area}) \div 10 = \frac{\pi r^2}{10} = \frac{\left(\frac{22}{7} \times 17.5 \times 17.5\right)}{10} = \frac{962.5}{10} = 96.25 \text{ cm}^2$ . (Equivalently, using $\theta = 36°: \left(\frac{36}{360}\right) \times \pi r^2 = 96.25 \text{ cm}^2$ .)

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

Marking Scheme

11 mark: correct area of the full circle $\pi r^2 = 962.5 \text{ cm}^2$ , or correct sector angle $\theta = 36°$ with the sector formula $(\theta/360)\pi r^2$ .
21 mark: correct $\text{area of each sector} = 96.25 \text{ cm}^2$ with unit.
3Accept dividing total area by 10, or using $\theta = 36°$ — both must give $96.25 \text{ cm}^2$ .
4Only 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 $\theta = 36°$ in $(\theta/360)\pi r^2$ ).

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.

Analysis & 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 $360°/10 = 36°$ .
$\text{Area of each sector} = \left(\frac{\theta}{360}\right) \times \pi r^2$ , or equivalently the whole circle's area divided by 10.

Common mistakes

The '5 vs 10' confusion — using $72°$ instead of $36°$ , which doubles the true sector area.
Forgetting to divide by the number of sectors and reporting the full circle area ( $962.5 \text{ cm}^2$ ) as the answer.
Using diameter instead of radius inside $\pi r^2$ , 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 ( $10 \times 96.25 = 962.5 \text{ cm}^2$ ).

Common Mistakes

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

Interesting Facts

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

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

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

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

Frequently Asked Questions

What is the central angle of each sector?

5 diameters create 10 equal sectors, so each central angle is $360°/10 = 36°$ . A common slip is to write $72°$ 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: $\pi r^2/10 = 962.5/10 = 96.25 \text{ cm}^2$ . This avoids the $(\theta/360)\pi r^2$ formula, though that gives the same 96.25 cm².