hard formulas that involves PI?

Question

what are some "hard" or "complicated" formulas that involves PI.
please do not give me the area of a cirle or the circumference of a circle :)

thanks!

Answer 1

I don't know if I'd really call these "hard" or "complicated," but they are a step above simple area and circumference.



Surface area of a cylinder: 2(pi)rh + 2(pi)r^2

Volume of a cylinder: (pi)r^2h;

Surface area of a cone: (pi)rl + (pi)r^2

Volume of a cone: [(pi)r^2h]/3

Surface area of a sphere: 4(pi)r^2

Volume of a sphere: [4(pi)r^3]/3



In the above equations, r = radius; h = height; l = slant height



Maybe someone else can provide some trig formulas for you, since my trig is pretty rusty.

Answer 2

e^(i*t) = cos t + i*sin t



where



the imaginary number i is the square root of -1, the variable t is an angle in radians, and e is the sume from 0 to infinity of 1/n!.



from this you can deduce an identity involving pi.



let t =pi



then



e^(i*pi) = cos(pi) + i*sin(pi)



e^(i*pi) = -1 + 0



e^(i*pi) + 1 = 0



this is Euler's Identity and is really an extraodinary equation as it relates 4 extremely important numbers in mathematics with the imaginary number i.

Answer 3

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 . . . (my favourite)

Answer 4

Normal distribution:



e^(-(x-mu)^2/(2*sigma^2)) / (sigma * sqrt(2*pi))



where mu is the mean and sigma is the standard deviation

Answer 5

chord length = 2r·sin(θ/2)

θ must be in radians, not degrees.

Answer 6

Does this help? :-)