# Math Help - [SOLVED] calculating the length of a shadow

1. ## [SOLVED] calculating the length of a shadow

How does one calculate the lenth of a shadow at any given time
lets say my latitute is 36 degrees.

And the height of the object is 1.2 meteres

also how do i claculate when the length of the shadow is the same as the height thats is 1.2 meters

Thanks Rod

2. A real calculation of the length of shadow is non-trivial and is in the domain of celestial navigation. See here for calculators for educational purposes.

3. Originally Posted by romed
How does one calculate the lenth of a shadow at any given time
lets say my latitute is 36 degrees.
And the height of the object is 1.2 meteres
also how do i claculate when the length of the shadow is the same as the height thats is 1.2 meters
Thanks Rod
Hello,

I can give you only a few informations:

1. When the sun is perpendicular over the tropic it has it's maximum altitude over the horizon at 12 o'clock local time . At a latitude of 36°N the maximum altitude of the sun is approximately 77.5° over the horizon. Then you get the shortest shadow:
$s=1.2 m \cdot \cot(77.5^\circ) \approx 26.6 cm$

The longest shadow you get when the sun touches the horizon: The shadow has an infinite length.

2. When the shadow is as long as the stick itself, then the sun has a altitude of 45° over the horizon. This situation will happen twice a day in the summer.
In winter the maximum altitude of the sun reaches only approximately 30.5° (at 36°N, of course!). That means the shortest shadow will have a length of

$s=1.2 m \cdot \cot(30.5^\circ) \approx 2.04 m$

I hope these informations are of some use to you.

Greetings

EB

4. Originally Posted by romed
How does one calculate the lenth of a shadow at any given time
lets say my latitute is 36 degrees....

Thanks Rod
Hello,

it's me again.

I've attached a diagram, to demonstrate where all my numbers came from.

Greetings

EB

5. Originally Posted by earboth
Hello,

I can give you only a few informations:

1. When the sun is perpendicular over the tropic it has it's maximum altitude over the horizon at 12 o'clock local time . At a latitude of 36°N the maximum altitude of the sun is approximately 81° over the horizon. Then you get the shortest shadow:
$s=1.2 m \cdot \cot(81^\circ) \approx 19 cm$

The longest shadow you get when the sun touches the horizon: The shadow has an infinite length.

2. When the shadow is as long as the stick itself, then the sun has a altitude of 45° over the horizon. This situation will happen twice a day in the summer.
In winter the maximum altitude of the sun reaches only approximately 27° (at 36°N, of course!). That means the shortest shadow will have a length of

$s=1.2 m \cdot \cot(27^\circ) \approx 2.36 m$

I hope these informations are of some use to you.

Greetings

EB
Hi. According to this page from NASA, the maximum and minimum altitudes of the sun at latitude $L$ are given by $90 - L \pm 23.5,$ which for $L = 36$ yields 77.5 and 30.5. How did you calculate your maximum and minimum?

6. Originally Posted by JakeD
Hi. According to this page from NASA, the maximum and minimum altitudes of the sun at latitude $L$ are given by $90 - L \pm 23.5,$ which for $L = 36$ yields 77.5 and 30.5. How did you calculate your maximum and minimum?
Hello,

you are right. Instead of 23.5° I took 27°. I can only guess why I use this number: The polar circle has a latitude of nearly 67° (and I'm living nearer to the polar circle than to the 36°N regions) and so I mixed up those ciphers. So sorry!

Greetings

EB