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How to Measure Shadows to Find "Similar Triangles" and Calculate the Height of an Object
Posted: 20 September 2014

How do you find the measurement of something when it is too tall to measure? One of the known techniques is called “Similar Triangles” which can be useful for solving problems involving heights too high to measure.

To Measure the Height of The Cactus only Measure the Shadow
There is a way that you can measure the height of cactus without touching it that would be more accurate than using a tape measure anyway.  

The shadows that are made by the cactus on the ground can be used in a similar triangle formula to find the height of the cactus. If you want to measure the tallest cactus, it's shadow would be the equivalent of the adjacent side in a diagram of a right triangle. It's height would be equivalent to what's called "the opposite side length" in math. The opposite side is always the "vertical" side of the triangle and the adjacent side is always the horizontal side of the triangle. This particular cactus can potentially be measured with a tape measure, but other objects like buildings would not be within the range of things able to be measured except with a mathematical technique such as similar triangles. To compare the shadow of the cactus to something you will need to measure the shadow of some other smaller vertical object not pictured, like a short stick positioned upright between two rocks.

The way that you find the height of the cactus is by first measuring the shadow that it makes on the ground, which depending on the time of day will vary in length. This step must be accomplished during daylight hours when the sun is at least a certain level above the horizon either after sunrise or before sunset. At the same time that you take the measurement of the cactus’ shadow, you have to measure the shadow of the smaller vertical object, like the stick propped up vertically with its base touching the ground between two rocks or anything significantly shorter than the cactus that’s length and shadow can be easily measured.

right triangle
Above: Right triangle: the shadows are represented by the (horizontal) adjacent side and the cactus is represented by the (vertical) opposite side of the triangle. These two side lengths are related by the hypotenuse.

With the measurements of the shadows of the objects taken at the same time, and also the measurement of the smaller vertical object, you can find the height of the cactus.

Height of cactus = ?
Length of shadow of cactus = 32 inches
Height of stick = 13 inches
Length of shadow of stick = 7 inches

Step 1: Make a ratio of the relative quantities such that length of shadow is over height of object and set the ratios equal to each other.

32 inches
 
7 inches

=

(height of cactus)
 
13 inches

The shadow of the object is not longer than the object, so the domain of the solution to this problem should be more than 32 inches long, the length of the cactus' shadow.

Step 2: Cross multiply to Solve the Similar Triangle.

32 x 13 = 7 x (height of cactus)

416 = 7 x (height of cactus)

Step 3: Divide 416 by 7 to find the height of the cactus

416 divided by 7 = 59.42857143

Step 4: Convert inches to feet:

59.42857143 inches divided by 12 inches = 4.952380952 feet

Step 5: Round off to nearest whole number. According to the analytical approach of measuring the cactus with similar triangles without touching it, the cactus is almost 5-feet tall.

 

 
By Andi Boggs