Sun angle and shadow length
A shadow's length is the object's height divided by the tangent of the Sun's angle above the horizon. At 45° a shadow is as long as the object is tall; at 30° it is 1.73 times as long; at 60°, 0.58 times. Below: a calculator for any place, date and time, the noon Sun and its shadow month by month at 40° north, hour by hour on the June solstice, and the noon Sun by latitude.
Sun angle and shadow length calculator
Enter your latitude and longitude (west is negative), the time zone your clock is on that day, a date and time, and the height of the object. The shadow comes out in the same unit as the height.
Sun angle above the horizon
The date starts at today; without JavaScript, the example above is a 6 ft person in New York at 3 p.m. daylight time on June 21, 2027. Below 5° the shadow length is not given.
How to calculate shadow length
Shadow length = height ÷ tan(Sun angle). The Sun angle, or solar elevation, is measured from the horizon up to the Sun; NOAA describes elevation as measured vertically from the point on the horizon below an object up to the object. The lower the Sun, the faster the shadow grows: from 30° to 15° a shadow doubles and more.
It works backward, too: Sun angle = arctan(height ÷ shadow length). A 6 ft person casting a 4 ft shadow means the Sun is 56.3° up.
A shadow points directly away from the Sun, at the Sun's azimuth plus 180°. NOAA measures azimuth clockwise from true north, so 90° is east and 180° is south.
The noon Sun through the year (40° north)
At 40° north, a shadow at noon is 0.30× the object's height in mid-June and 1.98× in mid-December. The times are by the clock on your time zone's meridian, with daylight saving time from March to early November.
| Month | Solar noon | Sun angle | Shadow |
|---|---|---|---|
| January | 12:09 p.m. | 29.0° | 1.81× |
| February | 12:14 p.m. | 37.4° | 1.31× |
| March | 1:09 p.m. | 48.0° | 0.90× |
| April | 1:00 p.m. | 59.9° | 0.58× |
| May | 12:56 p.m. | 68.9° | 0.39× |
| June | 1:01 p.m. | 73.3° | 0.30× |
| July | 1:06 p.m. | 71.5° | 0.33× |
| August | 1:05 p.m. | 64.0° | 0.49× |
| September | 12:55 p.m. | 52.9° | 0.76× |
| October | 12:46 p.m. | 41.4° | 1.13× |
| November | 11:45 a.m. | 31.5° | 1.63× |
| December | 11:55 a.m. | 26.8° | 1.98× |
Hour by hour on the June solstice (40° north)
On June 21, the Sun rises in the east-northeast and sets in the west-northwest; shadows swing the other way, from west-southwest in the morning to east-southeast in the evening. The shortest shadow, about 0.30× the height, falls a little after 1 p.m. daylight time.
| Time | Sun angle | Shadow | Shadow points |
|---|---|---|---|
| 6:00 a.m. | 4.1° | — | west-southwest (242.4°) |
| 7:00 a.m. | 14.6° | 3.85× | west-southwest (251.4°) |
| 8:00 a.m. | 25.6° | 2.08× | west (260.0°) |
| 9:00 a.m. | 37.1° | 1.32× | west (269.0°) |
| 10:00 a.m. | 48.5° | 0.88× | west (279.5°) |
| 11:00 a.m. | 59.5° | 0.59× | west-northwest (293.7°) |
| 12:00 p.m. | 68.9° | 0.39× | northwest (317.2°) |
| 1:00 p.m. | 73.4° | 0.30× | north (358.5°) |
| 2:00 p.m. | 69.4° | 0.38× | northeast (40.9°) |
| 3:00 p.m. | 60.1° | 0.57× | east-northeast (65.3°) |
| 4:00 p.m. | 49.2° | 0.86× | east (79.8°) |
| 5:00 p.m. | 37.8° | 1.29× | east (90.4°) |
| 6:00 p.m. | 26.3° | 2.02× | east (99.5°) |
| 7:00 p.m. | 15.2° | 3.68× | east-southeast (108.1°) |
| 8:00 p.m. | 4.7° | — | east-southeast (117.0°) |
The noon Sun by latitude
At the equinoxes the noon Sun stands about 90° minus your latitude; on the June solstice it is about 23.4° higher, on the December solstice about 23.4° lower. At 25° north the June noon Sun is 88.4° up, nearly overhead; at 50° north the December noon Sun is only 16.6°.
| Latitude | March equinox | June solstice | December solstice |
|---|---|---|---|
| 25° | 65.0° | 88.4° | 41.6° |
| 30° | 60.0° | 83.4° | 36.6° |
| 35° | 55.0° | 78.4° | 31.6° |
| 40° | 50.0° | 73.4° | 26.6° |
| 45° | 45.0° | 68.4° | 21.6° |
| 50° | 40.0° | 63.4° | 16.6° |
Why no length below 5°
Near the horizon, a hundredth of a degree changes a shadow a lot: at 1° up, a 1 ft post casts a 57 ft shadow, and the ground is never flat enough for that number to mean much. Below 5° this page gives only the direction.
Where Angel Step fits
Angel Step reads only your step count from Apple Health and shows it as one number on a living sky. It does not work out the Sun's position; that is this page's job. Your own shadow is the easiest sundial there is, and a walk is a good time to look at it.
Sources
- NOAA Global Monitoring Laboratory — solar calculator glossary: azimuth measured clockwise from true north, elevation measured up from the horizon. NOAA
- 15 U.S.C. § 260a — daylight saving time, used for the clock times. Cornell LII
- National Astronomical Observatory of Japan — sun altitude and azimuth service, against which the same code is checked for Japan, Korea and Taiwan. NAOJ
Updated