how do you calculate height using an objects shadow?

Answer 1

The simplest arrangement would be if, when you stood at the end of the object's shadow, there was a 45-degree angle up to the top of the object. In that case, the height and the length of the shadow would be identical. However, it's unlikely that you'll find that particular situation, so here's what you can do ...
Imagine that you have a triangle, which is made up of (a) the Perpendicular - which is the upright side of the triangle, i.e. the object we want to measure; (b) the Base - which is the length of the shadow, and (c) the Hypotenuse - which is the sloping side of the triangle, i.e. the (imaginary) line from the top of the object to the end of the shadow.
This triangle has an angle at its three vertices, or "corners". The one at the bottom of the object is a right-angle (90 degrees). What we need to do is to measure the angle between the shadow along the ground and the line of the Hypotenuse up to the top of the object. Obviously the taller the object is, the greater the angle will be. Using the rules of trigonometry, the tangent of that angle, A, is the Perpendicular, P, divided by the length of the Base, B; in other words, the height of the object divided by the length of the shadow. So Tan(A) = P/B. We can rearrange this to give Tan(A) x B = P. Suppose angle A was 60 degrees, then Tan(A) is 1.7 and if the shadow was 50 metres long, then the value of P is 1.7 x 50 = 86.6 metres high.

2,300 years ago, Eratosthenes used something similar to calculate the size of the Earth. He discovered that when the sun was overhead at Syene, it cast a shadow at Alexandria of 1/50 of a circle. He reasoned that this was also the angle at the centre of the Earth between Syene and Alexandria, and was an angular representation of the distance between the two places. Therefore if he knew the actual distance, this would be 1/50 of the total circle which was the distance all the way around the Earth. So he arranged for a troop of soldiers from the Roman army to pace off the distance, which turned out to be around 500 miles. Multiplying this by 50 gave a value of 25,000 miles for the circumference of the Earth.

We can do something similar today. Suppose the sun is overhead on the Equator, as it is at the time of the Spring and Autumn Equinox, and also that you know the distance from your position to the Equator. All you need to know now is the angle from your position to the Equator. You can find this by calculating the angle of a shadow cast by the sun when it is at its highest. How do you do this? By reversing the calculation above. Hold a 1-metre rule so that it just touches the ground (giving the Perpendicular, P) and measure the length of the shadow, which will be the base of the triangle, (B). This time we are interested in the angle at the top of the triangle. The tangent of this angle will be given by the Base divided by the Perpendicular, i.e. the length of the shadow (in centimetres) divided by 100 centimetres (which is the reason for using a 1-metre rule!). If the shadow is 128 cm long, dividing this by 100 gives 1.28, which is the tangent of the angle we require. The angle whose tangent is 1.28 (i.e. the ArcTan of 1.28) is 52 degrees. At 52 degrees north (or south) the distance to the Equator is 3,590 miles. Divide this by 52 and multiply by 360 and you have a value of 24,854 miles for the circumference of the Earth through the poles, which is a little less than the circumference around the Equator.
I have developed a spreadsheet which allows me to do this at any time of the year, not just at the Equinoxes. It takes into account the Equation of Time (which allows for the fact the that Earth does not move at a constant speed around the Sun), the longitude and latitude where the measurement is being taken, and it also allows me to do the measurement at any time of day, so it doesn't matter if it's cloudy when the sun is at its highest. If anyone is interested in me doing this at their school, then contact me.

Answer 2

Assuming the object is perpendicular to the ground:

You need to measure the length of the shadow and the angle of the light source at the end of the shadow. The height is then the length multiplied by the tangent of the angle.

Answer 3

Yes yes yea you could use trigonometry and multiple cosines and tangents etc but then you need to know the angle of elevation of the sun and to do that you need the latitude etc etc so the easy answer to this is carry round with you a stick of a known length i.e. 1 metre long or 1 ft or whatever depending on whether you're imperialistic or metrically inclined and then hold it upright and measure it's own shadow. Then go measure the shadow of the tree or whatever you're trying to discover the height of and then multiply that by the ratio of the stick height to the stick shadow.

Done and dusted.

Answer 4

The shadow isn't a results of the historic past, yet extremely a results of light. every time you're taking gentle and direct it at a concern, you will get a shadow with the aid of fact it won't be able to penetrate via opaque products (like human beings). you are going to ought to stability issues by employing employing one greater lights source (off digicam flash, yet another reflector, an open window, the solar, etc), having the priority step greater beneficial faraway from the backdrop so the shadow falls decrease to the floor instead of the wall, or changing the backdrop to a darker shade that isn't coach shadows. additionally, in case you're employing a flash you will in some situations get a very nasty "shadow halo" around the priority, which would be combatted by employing employing the flash off the digicam.

Answer 5

put your arm at 45degrees and use it to observe to top ot he item to be measured. start at the base of the item and walk backwards, when you sight the top, youre as far from the base,plus your own height . (45degrees is 1/1 gradient. for every foot out, its a foot up. pythagorus theorem... and its quite simple

Answer 6

It depends on the question and what is given.

If the item is on a hill, etc. you will have to make your own right triangle and use pytahorean theorem, law of sins or law of cosins.