How to calculate the height of a tower from it is shadow

Determining the height of a tower by measuring its shadow is a practical application of geometry, specifically the properties of similar triangles. When the sun shines on objects, it casts shadows that form right triangles, and the ratios of corresponding sides of these triangles are equal.

Step-by-Step Method

Understand the Concept of Similar Triangles The sun’s rays hit the tower and the reference object (e.g., a stick) at the same angle, creating two right triangles: one formed by the tower and its shadow, and another by the stick and its shadow. Because the triangles are similar, the ratios of their corresponding sides are equal. This gives us the proportion: $\frac{Height of tower}{Length of tower’s shadow} = \frac{Height of stick}{Length of stick’s shadow}$ $Length of tower’s shadow Height of tower = Length of stick’s shadow Height of stick$
Set Up the Reference Object
- Choose a stick or pole with a known height (e.g., 1 meter).
- Place it vertically in the ground so it stands upright and casts a clear shadow.
- Ensure the stick is on a flat surface near the tower to minimize differences in terrain.
Measure the Stick’s Shadow
- Use a measuring tape to measure the length of the stick’s shadow from the base of the stick to the tip of the shadow.
- Record this length (e.g., 1.5 meters).
Measure the Tower’s Shadow
- Measure the length of the tower’s shadow from the base of the tower to the tip of its shadow.
- Ensure you measure at the same time as the stick’s shadow to keep the sun’s angle consistent.
- Record this length (e.g., 30 meters).
Set Up the Proportion Using the similar triangles principle, set up the equation: $\frac{Height of tower}{Length of tower’s shadow} = \frac{Height of stick}{Length of stick’s shadow}$ $Length of tower’s shadow Height of tower = Length of stick’s shadow Height of stick$ Rearrange to solve for the tower’s height: $Height of tower = Height of stick \times \frac{Length of tower’s shadow}{Length of stick’s shadow}$ $Height of tower = Height of stick \times Length of stick’s shadow Length of tower’s shadow$
Calculate the Height Plug in the measured values. For example:
- Height of stick = 1 meter
- Length of stick’s shadow = 1.5 meters
- Length of tower’s shadow = 30 meters
$Height of tower = 1 \times \frac{30}{1.5} = 1 \times 20 = 20 meters$ $Height of tower = 1 \times 1.5 30 = 1 \times 20 = 20 meters$ So, the tower’s height is 20 meters.
Verify Your Measurements
- Double-check the shadow lengths to ensure accuracy.
- Measure multiple times if possible to confirm consistency.
- If the ground is uneven or the shadow is distorted, reposition the stick or adjust for inaccuracies.