How to Determine the Length of a Shadow

How to Determine the Length of a Shadow
••• AlexLinch/iStock/GettyImages

Determining the length of a small shadow on a surface is as easy as using a measuring tape or yardstick to measure the shadow. But for larger objects, such as a tall building, determining the length of the shadow is a little more difficult. It is not always practical to manually measure a shadow’s length. But if the height of the object casting the shadow you want to measure is known, you can use a formula to determine the length of the shadow. A shadow's length varies depending upon the angle of the light source.

    Visit the “Sun or Moon Altitude / Azimuth Table” Web page on the U.S. Naval Observatory website, or the “Altitude and Azimuth of the Sun” Web page on the website.

    Determine the altitude of the sun at the location of the shadow by using the calculator tool on this webpage. You need to know this figure to calculate the length of the object’s shadow. Using the U.S. Naval Observatory sun altitude calculator tool, you are asked to enter a date and the city and state where the object is located. Using the sun altitude calculator tool, you are asked to provide the same information, but instead of specifying the city and state, you must enter that latitude and longitude of the object’s location. For example, in Boston, Mass. on June 21, 2011, at noon the sun’s altitude is 70.9 degrees.

    Compute the tangent of the sun’s altitude angle in degrees (written “tan θ”). To perform this operation on a scientific calculator, enter the degree number then press the “Tan” button. (To turn your calculator that is installed in Windows into a scientific calculator, open the calculator, go to the “View” menu and select “Scientific.”) For example: tan(70.9) = 2.89

    Rewrite the following formula with the numerical equivalents: Object Height / tan θ = Shadow Length. For example, for the 790-foot high Prudential Tower in Boston, the formula would be 790 / 2.89 = Shadow Length.

    Calculate the formula to determine the shadow length. For example: 790 / 2.89 = 273.36 feet. Because the height of the Prudential tower was given in feet, the calculated length of the shadow is also in feet. In this example, it is learned that the length of the shadow cast by the 790-foot high Prudential Tower in Boston on June 21, 2011, at noon is approximately 273.36 feet.


    • To use this formula to determine the length of a shadow of an object not illuminated by sunlight, instead of performing Steps 1 and 2 to determine the altitude of the sun, you must determine the altitude of the light source that is illuminating the object.

Related Articles

What Is the Difference Between Umbra and Penumbra?
How to Calculate Height
How to Convert XY Coordinates to Longitude and Latitude
How to Use a Bushnell Reflector Telescope
How to Measure the Height of a Flagpole
How to Calculate Distance Angle
What Astronomical Instrument Measures the Brightness...
How to Calculate the Winter Solstice Sun Angle
How to Calculate Perimeter and Area Ratio
School Projects on Lunar Eclipses
How to Make a Simple Theodolite
How to Make a Model of the Solar System
How to Make a 3D Model of the Sun, Earth & Moon
How to Convert Lux to Candela
How to Find Mass in Weight
How to Calculate Angles Without a Protractor
Math Activities for the Outdoor Classroom
How to Calculate Epicenter
How to Measure a Magnification Mirror
How Much Time Is One Day on Mars?

Dont Go!

We Have More Great Sciencing Articles!