Measure the Earth like Eratosthenes

The question

Can I measure the size of the whole Earth with a stick and a tape measure?

Learn just enough first

Read the story of Eratosthenes and the well at Syene
Look up what "solar noon" is, and find its exact time for my town on the day

Kit

A straight stick or pole, about 1 m long, set exactly vertical (use a spirit level or plumb line)
Flat, level ground
Tape measure and a calculator

My prediction

The accepted circumference is about 40,000 km. How close will I get?

Method

Do this on or near the spring equinox (around 20 March), when the maths is simplest.
At solar noon, mark the tip of the shadow and measure the shadow length $s$ and stick height $h$.
Work out the Sun's angle from vertical: $\theta = \tan^{-1}(s/h)$.
At the equinox, the Sun is directly overhead at the equator, so $\theta$ equals my latitude — and my distance from the equator is easy to look up.

Data

Stick height $h$ (m)	Shadow length $s$ (m)	Angle $\theta$ (°)

Result

My angle $\theta$ is the slice of the full circle between me and the equator, so:

$$ C = \frac{360^\circ}{\theta} \times d_{\text{equator}} $$

where $d_{\text{equator}}$ is my distance due south to the equator. Even better: team up with a friend in another town north or south of me, measure on the same day, and use the difference in angles and the distance between you — just like Eratosthenes.

Explain it to a friend

Because the Earth is curved, the Sun shines at different angles in different places at the same moment. The difference in shadow angle tells you what fraction of the full circle lies between two places — measure the distance between them and you can scale up to the whole planet.

New questions

How much does a 1 cm error in the shadow change my answer?
How would I do this if I couldn't wait for the equinox?