(iii) (a) What is the total length of silver wire required ?

Areas Related to Circles

Total wire = circumference + 5 diameters

Official Answer

$\text{Total silver wire required} = \text{circumference} + \text{length of 5 diameters} = 110 + (5 \times 35) = 110 + 175 = 285 \text{ cm}$ . Both components (the rim and all five internal diameters) must be added to reach the final length.

total length of wirecircumference5 diameters110 + 175285 cm5 × 35brooch

Marking Scheme

11 mark: recognising $\text{total} = \text{circumference} + 5 \times \text{diameter}$ , i.e. adding the 5 diameters (175 cm) to the rim.
21 mark: correct arithmetic $110 + 175 = 285 \text{ cm}$ with unit.
3Accept full marks for any correct equivalent, e.g. $110 + 5(35) = 285 \text{ cm}$ .
4Only 1 mark if the 5 diameters are computed but the circumference is omitted, or vice versa.

Hint

Add the outer rim ( $\text{circumference} = 110 \text{ cm}$ ) to the wire for the 5 straight diameters ( $5 \times 35 \text{ cm}$ ).

Quick Oral Answer

The wire makes the outer circle, 110 centimetres, plus five diameters of 35 each, that is 175 centimetres, giving a total of 285 centimetres.

Analysis & Explanation

This 2-mark part is the synthesis step of the brooch case study, combining the rim and the internal spokes into one total length.

Concept

The wire is used twice: once for the circular rim (circumference) and once for the 5 internal diameters.
$\text{Total wire} = \text{circumference} + 5 \times \text{diameter} = 110 + 175 = 285 \text{ cm}$ .

Common mistakes

Reporting only the circumference (110 cm) and forgetting the spokes.
Using 5 radii (87.5 cm) instead of 5 full diameters (175 cm), missing that each internal wire crosses the whole circle through the centre.

Real-world

A craftsperson costing a wire-frame ornament must add the boundary length to every internal strut to know how much material to buy — precisely this calculation.

Common Mistakes

1Forgetting to add the 5 diameters and giving only the circumference (110 cm).
2Using 5 radii ( $5 \times 17.5 = 87.5$ ) instead of 5 diameters ( $5 \times 35 = 175$ ) — the wire runs fully across the circle, so each is a diameter.
3Adding only some of the diameters, or double-counting the centre, instead of taking 5 full diameters.

Interesting Facts

The 5 diameters all pass through one centre, and 5 diameters produce 10 equal sectors (each $360°/10 = 36°$ ) — the wire cleverly creates a 10-spoke pattern.

Total wire 285 cm is more than 2.5 times the rim length alone, showing the internal spokes use most of the silver.

Only 5 straight wires are needed for 10 divisions because each diameter serves two opposite sectors at once.

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Frequently Asked Questions

Why do we add 5 diameters and not 5 radii?

Each internal wire runs the full width of the circle, from one point on the rim through the centre to the opposite point — that is a diameter (35 cm), not a radius. So the five spokes total $5 \times 35 = 175 \text{ cm}$ .

Why add the circumference at all?

The single piece of silver wire forms both the outer circular boundary (the rim) and the internal spokes. The rim length is the circumference (110 cm), so the $\text{total wire} = \text{rim} + \text{spokes} = 110 + 175 = 285 \text{ cm}$ .