The sun on the hottest day of the year gets to just about 23 degrees latitude above the equator. Eretz Yisroel is about 34 degrees north. That means that the sun is always at least on an 11 degree angle coming down over Eretz Yisroel.
According to one shitta in the Gemara in Sukkah, a Sukkah that is more than 20 amos high is only pasul when the width and length are both just 7 tefachim - because then the shade is coming from the walls. But in a sukkah that is larger than 7x7, you can make it higher than 20 amos because the shade will come from the sechach.
But if you figure 11 degrees of angle on the sunlight, and to commutate using sin cos and tan, you will see that in a 7x7 sukkah the sunlight coming on 11 degrees will only make it about 8 amos down the wall, not nearly the 20 amos needed?
And that, my students, is why you need to know trigonometry.
ETA:
Nice cheshbon except that the Gemara actually says there IS NO צל סכך when it is more than 7x7 wide. What the Gemara actually says is that if the succah is more than ד׳ אמות wide it is ok even above כ׳ אמה according to one שיטה because then you will have צל סכך.
Also, keep in mind that the sun can come it at a diagonal from corner to corner as well, making the ratio approximately 5.6 Amos wide by 20 Amos tall.
Given these measurements there will be צל סכך for several weeks out of the year. So long as the succah is made in a way that provides צל at some point of the year that would suffice, as we find סוכת גנב״ך ורקב”ש is considered עשוי לשם צל.
And that, my friends, is why you should look things up in the Gemara.
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