Shadow Length Calculator

Introduction to Shadow Length and Solar Trigonometry

Have you ever wondered why your shadow looks extremely tall in the evening but almost disappears at noon? The length of a shadow is determined by the geographic relationship between an object, the ground, and the position of the sun. Using a shadow length calculator allows students, children, and scientists to compute the exact dimensions of a shadow cast by any object at any coordinates on Earth.

If you are interested in mapping sunlight for other purposes, you can use our Golden Hour Calculator to find the best photography times, or use the Sun Exposure Seat Finder to select the best shaded seat on a bus. For solar energy systems, our Solar Panel Angle Calculator calculates the optimal tilt angle based on your location's latitude.

The math that determines shadow length relies on solar elevation angle (altitude). The solar elevation angle measures how high the sun is relative to the horizon plane, ranging from 0 degrees (at sunrise or sunset) to 90 degrees (when the sun is directly overhead at the zenith). By applying basic right-angle trigonometry, we can convert the height of an object and the sun's angle into a precise horizontal measurement.

This tool processes coordinates, dates, and times entirely client-side using JavaScript. Your precise location and browser data remain completely private and are never transmitted to any external server.

Trigonometric Formula: How Shadow Size is Calculated

Calculations for a shadow length calculator are based on a right-angled triangle formed by the object (the vertical side), the ground shadow (the horizontal side), and the incoming solar rays (the hypotenuse).

Let's look at an example. If a child stands 150 cm tall and the solar elevation is 45 degrees, we calculate:

Shadow Length = 150 / tan(45°) = 150 / 1 = 150 cm

When the solar elevation is exactly 45 degrees, the shadow length is equal to the object's height. If the sun is higher in the sky (e.g. 60 degrees), the tangent value is greater than 1, making the shadow shorter than the object. If the sun is lower in the sky (e.g. 20 degrees), the tangent value is less than 1, making the shadow much longer than the height.

Step-by-Step: How to Use the Shadow Calculator

1. Select your visual character (Kid, Obelisk, or Tree) in the Visual Character card settings to adjust the 2D visual representation.
2. Choose your height unit: Metric for centimeters/meters or Imperial for feet and inches.
3. Input the height of the person or object in the height input fields.
4. Specify geographic coordinates by entering Latitude and Longitude manually, using the search address function, dragging the pin on the Leaflet map, or using the Locate Me device GPS option.
5. Adjust the date, timezone, and time slider to watch the sun orbit in real-time, observing the shadow length update on the dimension line.

Ancient Science: The Eratosthenes Earth Circumference Experiment

The study of shadows holds a historic place in scientific history. Around 240 BC, the Greek polymath Eratosthenes utilized vertical columns and solar shadows to calculate the circumference of the Earth with remarkable accuracy.

Eratosthenes heard reports that at solar noon on the summer solstice in the southern city of Syene (modern Aswan, Egypt), the sun shone directly down a deep water well, casting no shadows. This meant the sun was directly overhead at an elevation angle of 90 degrees.

Simultaneously in Alexandria, located directly north, he measured the shadow cast by a vertical pillar (obelisk) at solar noon. Using the height of the column and the shadow length, he calculated that the sun's rays hit the column at an angle of 7.2 degrees from the vertical.

Knowing that 7.2 degrees represents exactly 1/50th of a full 360-degree sphere, Eratosthenes reasoned that the distance between Alexandria and Syene must be 1/50th of the Earth's circumference. By multiplying the estimated distance by 50, he calculated a total circumference of approximately 250,000 stadia (about 40,000 kilometers), which is incredibly close to the actual modern value of 40,075 kilometers.

Educational Activities for Children and Students

Shadow calculations offer a hands-on, visual way to introduce children and physics students to astronomical mechanics and basic geometry:

• The Daily Shadow Journal: Stand outside at different hours (8:00 AM, 12:00 PM, 4:00 PM) and measure the actual length of your shadow. Input these times and coordinates into the calculator to see if your physical measurements match the trigonometric calculations.
• Equator Shadow Experiment: Set the latitude to 0 degrees (the Equator) and check different dates. Note how at the spring and autumn equinoxes, shadows disappear completely at solar noon, as the sun passes directly overhead.
• Finding the Height of a Tall Tree: You can measure the height of a tall tree without climbing it! Measure the tree's shadow on flat ground. Use the calculator to find the height-to-shadow ratio at the current time, or measure the shadow of a stick of known height at the same time to establish the ratio. Multiply the tree's shadow length by this ratio to find its height.