QUESTION: If I had an equalateral triangle with a 5" base, how tall is it to the top? Also do you know a good book about shapes and geometry a begginer could understand? I only took trig in college and that didn't really get into complex shapes or anything much beyond triangles.
ANSWER: Hi James, I will be happy to help.
Because an equilateral triangle has all 60 degree angles, an equilateral triangle can be split into two 30-60-90 triangles with the leg across from the 30 degree angle being half the side length, or 2.5" in our case, and the leg across from the 60 degree angle being the altitude we are trying to find.
Remember that 30-60-90 triangles have side length ratios, respectively, of x-xroot3-2x
Here we know our 2x is the side length 5, so x, as stated before, is 5/2 or 2.5
The altitude is xroot3 or 5*sqrt(3)/2
Any high school geometry book would cover triangles and polygons.
As to a book on shapes alone, I'm not sure, you may want to send that question to the question bank. Maybe someone else would know a good book.
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QUESTION: So you're saying that since the base is 5" I would take half of that, 2.5", and multiply that by the square root of 3, and that should give me the length from the middle of the base to the opposing top?
As far as a book of shapes I am particularly interested in dimensions in general and for example the question of what is the smallest number of X-sides shapes (of equal lengthed sides) will it require at minimum to create a ball. I think soccer balls use both pentagons and hexagons of certain size to complete that ball. In college I took a trig class, but we didn't get into complex shapes all. It feels to me to be a lost art. lol.
Yes, that is the length from the midpoint of one side to the vertex.
As far as books, I would suggest
Euler's Gem: The Polyhedron Formula and the Birth of Topology
David S. Richeson (Author)
Peter R. Cromwell (Author)