To determine the interval between segments when using systematic sampling, you divide the total population size by the sample size. We have a population of 800 students and want to select a sample of 40.

The interval, $k$, can be calculated using the formula:
$$k = \frac{\text{Population Size}}{\text{Sample Size}} = \frac{800}{40}$$

Performing the division:
$$k = \frac{800}{40} = 20$$

So, the interval between segments for the systematic sampling of 800 students with a sample size of 40 is $\boxed{20}$.