The spider's web

On a cold morning the garden fence is hung with webs, each one covered in tiny drops of dew. A spider has eight legs, not six like an insect, and she builds her web with them in the dark.

A cold morning: dew on the webs. A spider has $8$ legs, an insect only $6$.

legs: $8>6$.

She starts with straight threads from the centre, like the spokes of a wheel, twenty-four of them. They are spread evenly all the way round, so the angle between two neighbours is three hundred and sixty degrees divided by twenty-four, which is fifteen degrees.

$24$ spokes spread evenly, so the angle between neighbours is $360^{\circ }÷24=15^{\circ }$.

$360^{\circ }÷24=15^{\circ }$.

Then she spins one long thread in a spiral round the spokes. Say the web has thirty turns, each two centimetres further out than the last, so the turns have radii two, four, six and so on up to sixty centimetres. A circle has a length of two pi times its radius.

The spiral has $30$ turns, each $2\mathrm{cm}$ further out: radii $2,4,6,\dots ,60\mathrm{cm}$. A circle of radius $r$ has length $2\pi r$.

$r_k=2k\mathrm{cm},k=1,\dots ,30$; $C=2\pi r$.

Add the thirty circles: two pi times two times the sum of one to thirty, and that sum is four hundred and sixty-five. The total is about fifty-eight metres of thread, in a web only a little more than a metre wide.

Total: ${\sum }_{k=1}^{30}2\pi (2k)=4\pi \times 465\approx 5843\mathrm{cm}$, about $58\mathrm{m}$ of thread in a web barely $1.2\mathrm{m}$ wide.

${\sum }_{k=1}^{30}2\pi r_k=4\pi \times 465\mathrm{cm}\approx 58\mathrm{m}$.