What is the circumference of the brooch ?

Areas Related to Circles

Areas Related to Circles — Case Study (Silver Brooch)

Official Answer

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

circumferencecirclebroochpi22/7diameter110 cm

Marking Scheme

11 mark: correct formula C = πd (or 2πr) with π = 22/7 correctly substituted, and correct final value 110 cm with unit.

Hint

Use $C = \pi d$ with $\pi = \frac{22}{7}$ (since 35 is a multiple of 7) for an exact answer — do not use 3.14.

Quick Oral Answer

Circumference equals π times the diameter; since 35 is a multiple of 7, using $\pi = \frac{22}{7}$ gives $\frac{22}{7} \times 35 = 110 \text{ cm}$ exactly.

Analysis & Explanation

Find the circumference of the brooch using $C = \pi d$ .

Concept

Circumference formula: $C = \pi d$ (equivalently $2\pi r$ ).
Diameter = 35 cm is a multiple of 7, so $\pi = \frac{22}{7}$ gives an exact answer.

Key points

$C = \left(\frac{22}{7}\right) \times 35 = 22 \times 5 = 110 \text{ cm}$ — no rounding needed.
This length represents only the outer rim wire, not the 5 internal diameter wires.

Common mistakes

Using $\pi = 3.14$ instead of 22/7, giving an inexact 109.9 cm rather than the intended exact 110 cm.

Common Mistakes

1Using $\pi = 3.14$ instead of $\frac{22}{7}$ , giving an inexact decimal answer $(\approx 109.9 \text{ cm})$ instead of the clean intended value of 110 cm.
2Using the radius (17.5 cm) in the formula $C = \pi d$ instead of $C = 2\pi r$ , i.e., forgetting to double the radius or mistakenly using d twice.
3Arithmetic slip while cancelling $\frac{35}{7} = 5$ .

Interesting Facts

The circumference (110 cm) represents the exact length of silver wire that would be needed just for the outer rim of the brooch, excluding the wire used for the internal diameters.

The constant π was first approximated as 22/7 by the Greek mathematician Archimedes around 250 BCE, and it remains the standard fraction used in Indian school mathematics for exact-answer problems.

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Frequently Asked Questions

Why use $\frac{22}{7}$ instead of 3.14 for pi here?

Because the diameter (35 cm) is a multiple of 7, using $\pi = \frac{22}{7}$ cancels exactly to give a clean whole-number answer (110 cm), which is the standard CBSE convention for such values.

Does the circumference include the 5 internal diameters of wire?

No — circumference refers only to the length of the outer circular boundary of the brooch, not the 5 internal diameter wires that create the 10 sectors.