Windy wrote:Why is the surface area for a pyramid such a pain in the ass to calculate?
Mainly because one of the key lengths you need - the distance from the tip to the centre of the base of each triangular face - isn't one of the standard measurements of a pyramid. Also partly because there's three different shapes of face, and there aren't even the same number of each.
Personally, I wouldn't bother with combining all the steps into one formula:
I'd calculate the 'height' of each triangular face (ie. how high it would be if it was tilted to be vertical): Sqrt(h^2 + (l/2)^2) for the 'side' triangle, by Pythagoras' theorem.
Meaning the area of that triangle is [that value] * 1/2 * w, by the usual triangle area formula
Reverse length and width and repeat to get the 'end' triangle.
Add those two areas together, and multiply by two because there's two of each kind of triangle.
Then add l * w for the rectangular base.
So sin and cos are trigonometry things related to the angles of a triangle and the ratio of their sides. So how come I keep seeing them in graphs and things completely unrelated to triangles?
Sines and cosines are properties of angles, whether that angle's in a triangle or not. They're usually defined in terms of right-angled triangles, but they have all sorts of other uses than simply measuring triangles. They're very prettily related to circles, for instance:
That's a consequence of how triangles turn up in all sorts of places - anywhere you have a line at an angle, you have a triangle, if you draw in the other two sides. It's very often useful to describe things in terms of horizontal and vertical, and you need sines and cosines to go between description in terms of length and direction, and in terms of horizontal and vertical distance. So anything that uses vectors and coordinate systems is probably going to have trig functions turn up somewhere.
You can
make any function that repeats itself - or any function if you only consider it on a finite domain - out of an infinite sum of sine waves, and that can be quite useful. In physics, you can
model any periodic phenomenon using sines, and indeed many things can be modelled using just one sine wave, because it happens that that kind of motion turns up a lot in the real world.
They're also related to
exponentials and complex numbers so you'll see them in that respect.
And they have quite interesting properties when you differentiate them, so you'll see them a lot in calculus.
I could go on. Sines and cosines get everywhere, they're not just for measuring triangles.