Shadow Length Calculator

About the Shadow Length Calculator

Calculates the length of a shadow cast by an object from its height and the sun's elevation angle above the horizon.

How to use this calculator

1. Enter the object height in your chosen units, such as meters or feet.
2. Enter the Sun elevation angle in degrees, which is the angle above the horizon.
3. Click calculate to find the shadow length.
4. Check the step-by-step result to see how the formula was used.

The formula explained

The formula computes the shadow length from a right triangle. It uses the object height and the tangent of the Sun’s elevation angle, so a larger angle gives a shorter shadow.

• shadow = the length of the shadow
• height = the height of the object
• elevation = the Sun’s elevation angle above the horizon

Step by step method

1. Write the formula as \(shadow = \frac{height}{\tan(elevation)}\).
2. Substitute the given height and angle into the formula.
3. Evaluate \(\tan(elevation)\), then divide the height by that value.
4. State the shadow length in the same units as the height.

Worked example

Problem. A pole is \(6\) meters tall, and the Sun’s elevation angle is \(30^\circ\). Find the shadow length.

1. Use the formula \(shadow = \frac{height}{\tan(elevation)}\).
2. Substitute the values, \(shadow = \frac{6}{\tan(30^\circ)}\). Since \(\tan(30^\circ) \approx 0.577\), compute \(6 \div 0.577\).
3. The shadow length is about \(10.4\) meters.

Tips and common mistakes

• Make sure the angle is the Sun’s elevation above the horizon, not the angle from straight up.
• Use the same units for the final answer as the height, because the calculator does not change units.

Frequently asked questions