How high does the Sun climb at noon? It depends on two numbers you already know: your latitude and the Sun's declination that day. Working with them means adding and subtracting signed numbers, as in adding and subtracting rational numbers, since south latitudes and southern declinations count as negative.
Why it works
Picture the meridian, the line from the south point on the horizon through the zenith to the north. The celestial equator crosses it at an altitude of $90^\circ - \text{latitude}$ in the south (the north celestial pole is $\text{latitude}$ up in the north, and the equator is $90^\circ$ from the pole). At noon the Sun is on the meridian, $\delta$ degrees north of the celestial equator. So its altitude is $90^\circ - \text{latitude} + \delta$. In general, for any place and date, $h_{\text{noon}} = 90^\circ - |\text{latitude} - \delta|$ (count south as negative).
Key idea
Noon altitude of the Sun $= 90^\circ - \text{latitude} + \delta$ for places north of the overhead Sun; in general $90^\circ - |\text{latitude} - \delta|$. With $\delta$ between $-23.4^\circ$ and $+23.4^\circ$, the Sun can be straight overhead only between the tropics.
Worked examples
Worked example: Denver through the year
Denver is at latitude $40^\circ$ N. Find the noon Sun's altitude on an equinox, the June solstice and the December solstice.
- Equinox, $\delta = 0^\circ$: $90^\circ - 40^\circ + 0^\circ = 50^\circ$.
- June, $\delta = +23.4^\circ$: $50^\circ + 23.4^\circ = 73.4^\circ$.
- December, $\delta = -23.4^\circ$: $50^\circ - 23.4^\circ = 26.6^\circ$.
- The noon Sun swings $46.8^\circ$ (twice the tilt) between the solstices.
Answer: $50^\circ$, $73.4^\circ$ and $26.6^\circ$.
Worked example: Down under
Sydney is at latitude $34^\circ$ S. How high is the noon Sun on the December solstice?
- Use the general rule with south negative: latitude $= -34^\circ$, $\delta = -23.4^\circ$.
- $|-34 - (-23.4)| = 10.6^\circ$, so $h = 90^\circ - 10.6^\circ = 79.4^\circ$, in the northern sky.
- December is summer in Sydney, so a high Sun makes sense.
Answer: $79.4^\circ$.
Worked example: Measuring Earth
Around 240 BC, Eratosthenes knew that at noon on the June solstice the Sun was straight overhead at Syene, while at Alexandria, about 800 km north, it was $7.2^\circ$ from overhead. How big around is Earth?
- The Sun's rays are parallel, so the $7.2^\circ$ difference is the angle between the two cities at Earth's center.
- $7.2^\circ$ is $\frac{7.2}{360} = \frac{1}{50}$ of a circle.
- Circumference $\approx 50 \times 800$ km $= 40{,}000$ km.
Answer: About $40{,}000$ km, close to the true $40{,}075$ km.
Common mistake
Many people think the noon Sun is straight overhead everywhere, at least in summer. It never is outside the tropics. In Denver the June noon Sun reaches $73.4^\circ$; it is never at $90^\circ$, and your shadow never vanishes. Check: overhead means $h = 90^\circ$, which needs latitude $= \delta$, and $\delta$ never goes beyond $23.4^\circ$.