Above is a bitmap image of a measurement taken from Olympia to Augusta. This measurement was taken from an Albers Equal Area conic projection. As you can see, the the distance depicted here is ~2550 miles (~4100km). When this same measurement was taken prior to projecting the map the distance was only ~2530 miles(~4070km).

In the attribute table for U.S. cities, by sorting the columns in ascending or descending order, I determined several statistics for U.S. capital cities. The latitude and longitudes were determined using the information button. There are several cities that list -99 as the elevation. Perhaps this is an indication that there is a large change in relief within the city limits.

Greatest population in 1990: Pheonix, AZ (111°59'51.687"W 33°34'49.804"N)

Lowest population in 1990: Montpelier, VT (72°31'34.179"W 44°24'4.628"N)

Capital city with the highest elevation: Santa Fe, NM (106°7'41.936"W 35°47'36.573"N)

Below is a picture of ArcMap layout view of projected maps of the lower 48 and of New Mexico.

Below is another screen capture, this on of the world in Sinusoidal projection. This view was created via importing the appropriate data, then selecting the dataframe>properties>coordinate systems tab>then choosing from the available projections. This operation changed the appearance of the map, but does not appear to have altered the source data.