You have a cubiod of dimension 2.2m by 1.8m by 2.0m, you have a pole of length 3m. Will the pole fit into the cuboid?

The longest straight line within the cuboid is the diagonal from a corner to the corner diagonally from it (show diagram), call this D. So to find out whether the pole will fit, find this length, if it is longer than the pole, then the pole will fit. We need to find the length from a bottom corner, to the corner diagonally across and above it. This is clearly the hypotenuse of a right angled triangle, and from pythagoras' theorem we know that the square of the hypotenuse is equal to the sum of the squares of the other lenths. So, we must find the other lengths. One of them is clearly just the height of the cuboid (say this is 2.2m), call this H. The other length is the diagonal of the bottom rectangle which has dimension 1.8m by 2.0m, call this E. The diagonal is the hypotenuse of a right angled triangle with other lengths 1.8m and 2.0m, by pythagoras' theorem (a^2 = b^2 + c^2) we can say; E^2 = 1.8^2 +2.0^2 = 7.24 => E = sqrt(7.24) = 2.6907.... We now have the two other lengths of the original right angled triangle, H (2.2m) and E (2.6907...m). So again, by pythagoras, we can say; D^2 = H^2 + E^2 = 2.2^2 + 2.6907^2 = 12.08 => D = sqrt(12.08) = 3.476... We have found that the longest straight line within the cuboid is 3.476m. This is greater than 3m, therefore the pole will fit.