How many zeros of a random polynomial are real?

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve <!-- MATH $(1,\,t,\,\ldots \,,t^{n})$ --> projected onto the surface of the unit sphere, divided by . The probability density of the real zeros is proportional to how fast this curve is traced out.