Hong Life CalculatorsUnderstanding the noon sun elevation and solar declination calculator — concept and principle
This calculator gives two numbers for any latitude and date: the solar declination (the latitude where the Sun is directly overhead at noon) and the Sun's elevation above the horizon at local solar noon, its highest point of the day. People use it to estimate solar panel tilt, shadow lengths, sunlight in rooms and gardens, and in geography lessons on the seasons.
How it is calculated
- Declination: δ ≈ 23.44° × sin(360°/365 × (N − 81)), where N is the day of the year (1 January = 1). Day 81 is close to the March equinox, when δ = 0.
- Noon elevation: α = 90° − |φ − δ|, where φ is latitude (north positive, south negative).
The day number uses the year you enter (default 2026), so leap years are counted correctly.
The principle
Earth's axis is tilted about 23.44° to its orbit. As Earth goes around the Sun, the point directly beneath the Sun moves between 23.44° N (June solstice) and 23.44° S (December solstice), tracing roughly a sine wave over the year; that is the declination. At solar noon the Sun lies on your meridian. If it were overhead at latitude δ and you stand at latitude φ, the angle between the zenith and the Sun is simply the difference in latitude |φ − δ|, so its elevation is 90° minus that difference.
Worked example
- Inputs: latitude 37.57° (Seoul), 21 June, so N = 172.
- 360/365 × (172 − 81) = 89.75°, and sin 89.75° = 0.99999, so δ = 23.43978294°.
- |37.57 − 23.44| = 14.13°.
- Elevation = 90 − 14.13 = 75.86978294°.
- A 1 m pole would cast a noon shadow of 1 ÷ tan 75.87° ≈ 0.25 m.
| Latitude (°) | Noon elevation (°) |
|---|---|
| 9.39 | 75.95271706 |
| 18.8 | 85.34521706 |
| 28.2 | 85.26228294 |
| 37.6 | 75.86978294 |
| 47 | 66.47728294 |
| 56.4 | 57.08478294 |
| 75.1 | 38.29978294 |
How to read the result
For Seoul the noon elevation ranges from about 75.9° at the June solstice to 52.4° at the equinoxes and only about 29.0° at the December solstice, when a 1 m pole casts a shadow about 1.8 m long. If the elevation comes out negative, the Sun stays below the horizon all day (polar night). Within the tropics the Sun can be overhead (90°) on two days a year. A fixed solar panel facing south is often tilted near the latitude angle as a year-round compromise, though the best tilt depends on climate and local guidance.
Common mistakes
- Using clock noon instead of solar noon; in Korea solar noon is usually around 12:30.
- Entering southern latitudes as positive numbers.
- Confusing elevation (from the horizon) with zenith angle (from overhead); they add to 90°.
Frequently asked questions
Why is the hottest month not June if the Sun is highest then?
Land and oceans take weeks to warm up, so peak temperatures lag the solstice; in Korea the hottest period is usually late July to August.
What latitude has the Sun overhead at the June solstice?
The Tropic of Cancer, at about 23.44° N, which equals the axial tilt.
How do I get the shadow length?
Shadow length = object height ÷ tan(elevation).
Sources: J. Meeus, Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998, Chapter 25 · J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes, 4th ed., Wiley, 2013