(ii) What is the circumference of the brooch ?

Q: Should I use $\pi d$ or $2\pi r$ for the circumference?

Both give the same result because diameter $d = 2r$. With $d = 35 \text{ cm}, \pi d = \left(\frac{22}{7}\right)(35) = 110 \text{ cm}$; with $r = 17.5 \text{ cm}, 2\pi r = 2\left(\frac{22}{7}\right)(17.5) = 110 \text{ cm}$.

Areas Related to Circles

Circumference of a circle

Official Answer

$\text{Circumference} = \pi d = \left(\frac{22}{7}\right) \times 35 = 110 \text{ cm}$ .

circumferenceC = πd2πr110 cmπ = 22/7perimeter of circlebrooch

Marking Scheme

11 mark: correct circumference = 110 cm using $C = \pi d$ or $2\pi r$ with $\pi = \frac{22}{7}$ .
2Accept 110 cm; using $2\pi r$ with $r = 17.5 \text{ cm}$ must also give 110 cm.
3No marks for using area formula $\pi r^2$ instead of the perimeter formula.

Hint

Use $C = \pi d$ with $\pi = \frac{22}{7}$ ; the 35 cancels the 7 neatly.

Quick Oral Answer

Using C equals pi times diameter, that is 22 over 7 times 35, the seven cancels to give 22 times 5, which is 110 centimetres.

Analysis & Explanation

This part tests the circumference formula applied to the outer rim of the brooch.

Concept

$\text{Circumference of a circle} = \pi d = 2\pi r$ ; with $d = 35 \text{ cm}$ and $\pi = \frac{22}{7}$ , the 7 cancels neatly to give 110 cm.

Common mistakes

Reaching for πr² (an area formula) instead of the perimeter formula, giving the wrong quantity (962.5) altogether.
Substituting the diameter in place of the radius inside 2πr, which doubles the answer to 220.

Real-world

The circumference is the actual length of wire a jeweller would cut to trace the boundary of the brooch; an error here also carries forward into the total-wire calculation in the next part.

Common Mistakes

1Using the area formula $\pi r^2$ instead of the circumference formula $\pi d/2\pi r$ .
2Using $r = 35$ (diameter) inside $2\pi r$ , doubling the answer to 220 cm.
3Approximating $\pi$ as 3.14 and getting 109.9 cm when the intended clean answer with $\frac{22}{7}$ is exactly 110 cm.

Interesting Facts

Because $\pi = \frac{22}{7}$ and the diameter is $35 = 5 \times 7$ , the circumference comes out as the exact whole number 110 cm — the numbers are engineered for a clean answer.

Circumference is the amount of wire needed just for the outer rim of the brooch, before adding the 5 internal diameters.

$2\pi r$ and $\pi d$ are the same formula since $d = 2r$ — students may use whichever they find quicker.

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Frequently Asked Questions

Should I use $\pi d$ or $2\pi r$ for the circumference?

Both give the same result because diameter $d = 2r$ . With $d = 35 \text{ cm}, \pi d = \left(\frac{22}{7}\right)(35) = 110 \text{ cm}$ ; with $r = 17.5 \text{ cm}, 2\pi r = 2\left(\frac{22}{7}\right)(17.5) = 110 \text{ cm}$ .

Why does the answer come out exactly 110 cm?

Because the diameter 35 is a multiple of 7, the 7 in $\pi = \frac{22}{7}$ cancels: $\left(\frac{22}{7}\right) \times 35 = 22 \times 5 = 110$ . CBSE picks such numbers so the answer is a neat whole number.