During the "slice the cone" scene, the Spirit says, "A slice like this, and you have a searchlight. A slice like this, the mirror of a giant telescope." The actual cut made in the cone is a hyperbola, meanwhile both a searchlight and a telescope's mirror are both parabolas. (The difference is that a parabola is made by making an exactly vertical slice in the cone, not an angled slice as depicted.)
Despite this being a mathematical education film, a character
incorrectly recites the value of the mathematical constant pi. The character states, "Pi is equal to 3.141,592,653,589,747, et cetera, et cetera, et cetera." The correct value of pi (to the same number of digits) is actually 3.141,592,653,589,793. (The last two digits are different.)
During the "imagination" segment towards the end, the Spirit says to put a triangle in a perfect circle, and then turn the triangle. The image that results is a straight line that reaches both the top and bottom of the circle. There is no possible orientation that the original triangle could have to reproduce this visual effect in real life.