Sign of the time of the year

Happy Summer! This morning, the Sun reached its highest North declination and summer began! And fans of long daylight, panic! Because now the days start getting shorter.

Today around 1:15 PM, when the Sun reached the highest altitude(remember - daylight savings time!), I went outside, with a box, a compass with a ruler and a sheet from my notes book. What I was going to do? Measure the length of the box's shadow:

Unfortunately the cumulus clouds did not let me to take the reading at the exact moment when the Sun peaked over the horizon - this picture with the reading was done about 3-4 minutes later at 1:17 PM. Now what good I would know from the length of the shadow of a random object? This - the altitude of the Sun above the horizon in degrees. For those unaware about horizontal coordinates, here's a reference: If the object is at the horizon, it's altitude is zero degrees. When it is at zenith(right above your head - the Sun was/is visible there if you were/are at 23,44 North latitude at local noon today, you're yet to see it or are seeing it right now if you are in Americas and at that latitude), it's altitude is 90 degrees. But now again? What has the length of the shadow to do with the altitude of the Sun in the sky? Simple: It is just a matter of a trigonometry that every 9th Grader should know.

Basically, we can visualize the shadow, the object and the light beam as a triangle. Leaving the explanation of its shapes aside, we recall from school mathemathics(although I initially read about this trick from a Physics text book before learning about trigonometry when I was in 9th Grade) that tangent of angle alpha equals a divided by b.
tan alpha = a/b
Now, we got a as the height of the box, and b as the length of the shadow. We can get the angle alpha by using the arctangent function - so the height of the Sun equals arctangent of the height of the box divided by the length of the shadow.
height = arctangent(height of the box/length of the shadow)

Now: The box was 8,9 cm tall. The shadow was 6,2 cm long, and here is the result:
About 55,1 degrees above horizon! Now, because of the small dimensions of the object I used, one millimeter(accuracy of my measurement) in shadow size makes fairly big difference in the angle, and therefore this would have been much more accurate if the object used was taller, but you get the idea. Now, when it is measured at the moment the Sun reaches the highest altitude, it is possible to calculate the declination of the Sun(basically this is the latitude where people at their local noons are walking around, trying to see their shadows only to find them below their feet - the Sun is at zenith). How? You need to know your latitude for this. Mine is about 58,167 degrees North. I have the height of the Sun, 55,1 degrees. Now I will simply add my latitude to the height and subtract 90 degrees
decl = h+lat-90
decl = 55,1+58,167-90 ~ 23,27 ~ 23,3 degrees.
Now keep in mind about what I told about the accuracy before - my measurement in this test wasn't too accurate. As a matter of fact, had the length of the shadow been 6,1 cm, not 6,2 cm, the calculation would have yielded 23,7 degrees declination, far higher than the axial tilt of planet Earth(which is 23,44 degrees, and which also was Sun's declination at the moment of Summer Solstice). Plus, I made the measurement a few minutes after the Sun had peaked in its height.
Now, likewise, you can calculate the maximum height of the Sun from its declination:
hmax = decl + 90 - lat (if the height is higher than 90 degrees, it means the Sun will peak actually in the Northern sky, not Southern. In this case subtract the height from 180 degrees, like when hmax=111 then hmax=180-111)

So that's some math behind the Summer solstice! Enjoy the summer!