Elysian Shadows

So, my math teacher volunteered me for math field day, and I've been flipping through the packet, but I've found a problem that I can't seem to find out how to solve. I asked my calculus teacher and she didn't know either... So I thought perhaps some of you intelligent folks would know .

So here's the question in it's entirety:

"A square of side length 1 has equilateral triangles attached to the outside of each side. The total enclosed area can be written in the form (a + b * sqrt(c)) / d; where a,b,c,d are all relatively prime natural numbers. Find the sum a + b + c + d"

So, obviously the area is 3, but I'm not sure about how you can really use that to any advantage. I'm not really sure about the origin of that equation, but I assume it's somehow the area for a star? (Perhaps even just a 4 pointed star?) If that's the case, I would assume that one of those variables in some way describes the base, (which is a square in this case).

Anyways, I'd appreciate a point in the right direction, I googled this, but I really have no ideas, and I can't seem to find anyone who does. If you know what concepts this is referring to, go ahead and just link me, or just post the name of it and I can use google, I don't really care to get an answer from you guys, I just want to know how to do it

Well, just looking at a few things, A = 1 + sqrt(3). Why? Well, if you've taken geometry, the height of an equilateral triangle with base b is b/2*sqrt(3). With 4 equilateral triangles with bases of 1, the final total area comes out to sqrt(3). 1 is the area of the square. So, fitting it into your formula, a=1,b=1,c=3 and d=1, so the final answer is 6. And yes, a,b,c,d are all coprime

Ginto8 wrote:Well, just looking at a few things, A = 1 + sqrt(3). Why? Well, if you've taken geometry, the height of an equilateral triangle with base b is b/2*sqrt(3). With 4 equilateral triangles with bases of 1, the final total area comes out to sqrt(3). 1 is the area of the square. So, fitting it into your formula, a=1,b=1,c=3 and d=1, so the final answer is 6. And yes, a,b,c,d are all coprime

Wow... That was a lot simpler than I thought... I'm a dumb ass XD... For some reason I was thinking the height of the triangle was 1... Probably because each side was length 1, and ... Wow... Well, thanks XD

good job, I know you did it with the hight formula, but its one more step, and I think since they have the 4 eqi tris they want you to use the area formula instead since it works out better.

formula for area of a eqi tri, is A = (x^2*sqrt(3))/4 so when you have 4 of those, it becomes 4*(x^2*sqrt(3))/4, and thats where the 4 gets canceled out. thus ending up with x^2*sqrt(3)