Length of Shadow Calculator
What is the Length of Shadow?
The length of shadow is the distance from the base of an object to the tip of the shadow it casts on a surface due to a light source, typically the sun. This length depends directly on the height of the object and the angle at which the light rays strike the object relative to the surface. When the sun is high in the sky (large angle of elevation), the shadow is short. When the sun is low on the horizon (small angle of elevation), the shadow is long. Calculating the length of shadow is a practical application of basic trigonometry.
Anyone interested in photography, architecture, sundial design, or even just curious about the natural world can use a length of shadow calculator. For example, architects might use it to understand how shadows from a new building will affect surrounding areas at different times of the day and year. Photographers use it to plan shoots based on desired shadow effects. A common misconception is that shadow length is only about the time of day; while time of day influences the sun’s angle, the object’s height is equally crucial.
Length of Shadow Formula and Mathematical Explanation
The relationship between the object’s height, the sun’s angle of elevation, and the length of shadow can be visualized as a right-angled triangle:
- The object forms the vertical side (opposite to the sun’s angle from the ground at the tip of the shadow).
- The shadow forms the horizontal side (adjacent to the sun’s angle).
- The line from the top of the object to the tip of the shadow is the hypotenuse.
The sun’s angle of elevation (θ) is the angle between the ground and the line of sight to the sun.
The formula to find the length of shadow (L) given the object height (H) and the sun’s angle of elevation (θ) is derived from the tangent function in trigonometry:
tan(θ) = Opposite / Adjacent = H / L
Therefore, L = H / tan(θ)
Where:
- L = Length of Shadow
- H = Height of the Object
- θ = Angle of Elevation of the Sun (in degrees, but converted to radians for the tan function)
To use the `tan` function in most programming languages or calculators, the angle θ needs to be converted from degrees to radians: Angle in Radians = Angle in Degrees × (π / 180).
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| L | Length of Shadow | meters, feet, etc. | > 0 |
| H | Height of Object | meters, feet, etc. | > 0 |
| θ | Sun Angle of Elevation | degrees | 0 < θ < 90 |
| θ (radians) | Sun Angle in Radians | radians | 0 < θ < π/2 |
Practical Examples (Real-World Use Cases)
Example 1: A Tree’s Shadow
A tree is 15 meters tall. The sun is at an angle of 30 degrees above the horizon.
- Object Height (H) = 15 m
- Sun Angle (θ) = 30 degrees
- Angle in Radians = 30 * (π / 180) ≈ 0.5236 radians
- tan(30°) ≈ 0.57735
- Length of Shadow (L) = 15 / 0.57735 ≈ 25.98 meters
The tree will cast a shadow approximately 25.98 meters long.
Example 2: A Person’s Shadow
A person is 6 feet tall. The sun’s angle of elevation is 60 degrees.
- Object Height (H) = 6 ft
- Sun Angle (θ) = 60 degrees
- Angle in Radians = 60 * (π / 180) ≈ 1.0472 radians
- tan(60°) ≈ 1.73205
- Length of Shadow (L) = 6 / 1.73205 ≈ 3.46 feet
The person’s shadow will be about 3.46 feet long. Notice how a higher sun angle results in a shorter length of shadow compared to the first example.
How to Use This Length of Shadow Calculator
- Enter Object Height: Input the height of the object whose shadow you want to calculate.
- Enter Sun Angle of Elevation: Input the angle of the sun above the horizon in degrees. This angle must be between 0 and 90 (practically between 1 and 89.9 for this calculator).
- Select Height Unit: Choose the unit of measurement used for the object’s height. The shadow length will be calculated in the same unit.
- Calculate: Click the “Calculate” button or simply change the input values.
- View Results: The calculator will display the calculated length of shadow, the inputs used, and the formula. It also shows a table and a chart illustrating how the shadow length varies with the sun’s angle for the given object height.
- Copy Results: You can copy the main results and inputs using the “Copy Results” button.
The results give you the direct length of shadow on a flat surface perpendicular to the object’s base, assuming the ground is level.
Key Factors That Affect Length of Shadow Results
- Object Height: The taller the object, the longer its shadow will be for a given sun angle. A taller building will cast a longer length of shadow than a shorter one.
- Sun Angle of Elevation: This is the most dynamic factor. The lower the sun is in the sky (smaller angle), the longer the shadow. The higher the sun (larger angle), the shorter the shadow. This angle changes throughout the day and with the seasons.
- Time of Day: Directly influences the sun’s angle. Shadows are longest near sunrise and sunset and shortest around noon.
- Latitude: Your location on Earth affects the range of sun angles you experience throughout the year.
- Time of Year (Season): The Earth’s tilt causes the sun’s apparent path across the sky to change with the seasons, affecting the maximum and minimum sun angles during the day.
- Terrain/Surface: The calculator assumes a flat, horizontal surface. If the ground is sloped, the actual length of shadow will be different. A shadow cast uphill will be shorter, and downhill will be longer.
Frequently Asked Questions (FAQ)
- What happens to the length of shadow if the sun angle is 0 degrees?
- Theoretically, if the sun angle is exactly 0 degrees (at the horizon), the shadow length would be infinitely long as tan(0) = 0, and division by zero is undefined. Practically, it means a very long shadow as the sun rises or sets.
- What happens if the sun angle is 90 degrees?
- If the sun were directly overhead (90 degrees), tan(90) is undefined (approaches infinity), meaning the shadow length would be zero (H/infinity -> 0). The shadow would be directly beneath the object.
- Does the width of the object affect the length of its shadow?
- No, the width of the object does not affect the maximum length of shadow directly along the line from the sun, only the width of the shadow itself.
- Can I use this calculator for any light source?
- Yes, as long as you know the angle of elevation of the light source relative to the object and the surface, and the light source is far enough to be considered as casting parallel rays (like the sun).
- Why does the calculator have a limit of 1 to 89.9 degrees for the sun angle?
- Angles very close to 0 or 90 result in extreme values (very long or very short shadows) and can cause calculation issues or are less practical for typical shadow observations. 1 to 89.9 covers most realistic scenarios.
- How does latitude affect the length of shadow?
- Latitude determines the maximum and minimum sun elevation angles throughout the year and day, thus influencing the range of shadow lengths you’ll observe at a specific location.
- Is the Earth’s curvature considered?
- No, this calculator assumes a flat surface and does not account for the Earth’s curvature, which is negligible for most everyday objects and shadows.
- What if the ground is not flat?
- If the ground is sloped, the actual length of shadow will differ. This calculator assumes a flat horizontal surface onto which the shadow is cast.
Related Tools and Internal Resources
- Solar Angle Calculator: Calculate the sun’s position and angles for any location and time to get a more precise sun angle for the length of shadow calculation.
- Right Triangle Calculator: Explore more calculations related to right-angled triangles, which form the basis of shadow length calculations.
- Trigonometry Basics: Learn about the tangent function and other trigonometric concepts used in calculating the length of shadow.
- Sundial Design Principles: Understand how shadow lengths and angles are fundamental to designing sundials.
- Daylight Hours Calculator: See how daylight hours and sun position change throughout the year, affecting shadow lengths.
- Sun Position Algorithm: Delve into the details of calculating the sun’s position accurately.