Teaching · For Teachers · Earth and space science

How long is the shadow,
and which way
does it point?

Type an object’s height, a place and a time. The calculator finds the sun’s angle and bearing, then the shadow’s length and direction, and repeats it for every hour of daylight so a class can check it with a tape measure.

Find the shadowWhat you get

The sun and shadow calculator

The worked example is a 20 ft flagpole in Chicago on September 22. Nothing you type leaves this browser.

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What you get

Teaching · For TeachersFree

What you finish with
The sun’s elevation and bearing, the shadow’s length and direction, and an hour-by-hour table for the day. It prints as a summary and downloads as a spreadsheet.
Who it is for
Earth science, physics and math teachers and their students, and anyone checking where shade will fall.
Format
Browser calculator for a phone, tablet or computer, by touch or keyboard. Prints a summary; downloads a spreadsheet (.csv); export and import a file.
Access
Free. No account, no sign-in and nothing to install. Everything you type stays in this browser; nothing is sent to Caveat.
Where it lives
The For Teachers space in Caveat Teaching.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Measure the object’s height and choose the units.
  2. Type the place’s latitude, longitude and UTC offset.
  3. Choose the date and type the clock time.
  4. Read the shadow’s length and direction.
  5. Go outside at one of the hours in the table and measure it.

Find the shadow

How the shadow is worked out

The sun’s place in the sky, then one right triangle.

The worked example: a 20 ft flagpole in Chicago at 3:00 PMSun’s place38.1° upbearing 226° SWTangent0.784tan 38.1°Shadow25.5 ft20 ft ÷ tangentDirection46°opposite the sunThe worked example: a 20 ft flagpole in Chicago at 3:00 PMSun’s place38.1° upbearing 226° SWTangent0.784tan 38.1°Shadow25.5 ft20 ft ÷ tangentDirection46°opposite the sun
A shadow is a right triangle: the object is one side, the shadow the other, and the sun’s elevation is the angle at the shadow’s tip.

The sun’s place

The date gives the sun’s declination, how far north or south of the equator it is overhead. The clock time and longitude give the hour angle, how far the sun has turned past its noon position. With the latitude, those two angles fix the sun’s elevation above the horizon and its bearing from true north.

A class check

Measure a metre stick’s shadow at the times in the table and compare. Results within a few percent are normal; bigger gaps usually mean the ground is not level, the stick leans, or the UTC offset is off by the daylight saving hour. Ratios work in any units, so the shadow ÷ height column holds for every object at that moment.

Sources

  1. NOAA Global Monitoring Laboratory. Solar Calculator and its calculation details, including the refraction correction (gml.noaa.gov/grad/solcalc).
  2. Meeus, J. (1998). Astronomical Algorithms (2nd ed.). Willmann-Bell. Solar coordinates and the transformation to horizontal coordinates.
  3. National Renewable Energy Laboratory. Solar Position Algorithm (SPA) technical report (nrel.gov).
  4. NASA Science. Seasons, the sun’s path and Earth’s tilt (science.nasa.gov).

The worked example uses Chicago’s published coordinates; the flagpole is made up. Your entries stay in this browser. Support: team@caveatacademic.com.