Shadow Length Calculator

Trigonometry · Axiom Academy

LESSON Shadow Length Calculator Discover how the tangent function describes shadow length—and why shadows disappear when the sun is directly overhead. It's a beautiful day, and you notice your shadow changing length as the sun rises higher in the sky. Why does this happen? The answer lies in trigonometry! This 6-foot pole casts different shadow lengths depending on the sun's angle. Let's explore this relationship! Move the slider to change the sun's angle and watch what happens to the shadow! The angle is measured from the horizon. You may have noticed something strange as you moved the slider toward 90°. What happens to the shadow when the sun is directly overhead? At 90°, the shadow length becomes... The shadow length is directly related to the tangent function ! Here's why: At 90° (sun directly overhead): tan(90°) = height/0 = UNDEFINED! There is no shadow, so shadow length = 0. We cannot divide by zero, making tangent undefined. Tangent Relates Opposite to Adjacent tan(θ) = opposite/adjacent = height/shadow Shadow Length Changes with Sun Angle Lower sun → longer shadow; Higher sun → shorter shadow Tangent is Undefined at 90° When the sun is overhead, shadow length = 0, making tan(90°) = height/0 (undefined) Tangent Grows Without Bound As angle approaches 90°, tan(θ) → ∞ (shadow gets infinitely short) You've discovered how tangent describes real-world relationships—and why it has asymptotes at 90° and 270°.

This is the written version of the interactive lesson above. See the full Trigonometry course.