Bowls are the least efficient use of space, followed by cylinders, and the most efficient is square or rectangular. It's not magic or optical allusion - but as YT2095 pointed out - there are differences that can be deceptive when you just look at them, but can be easily explained by first semester Junior Hi/Middle School Algebra ... remember those old area and volume formulas you swore you would never need to know later in life?
1 Cup = 14.4375 cubic inches, 12 Cups/3 Quarts = 173.25 cu in. These are givens. Now, to get a 3 Quart capacity in various shaped containers:
Square: 8" W x 8" L x 2.71" H
Rectangle: 9" W x 13" L x 1.5" H
Round: 8" diameter by 3.5" high
Bowl: 8" diameter by 5-6" high (if the bowl was a true hemisphere it would be 8.8" diameter and 4.4 inches high).
Calculating the volume of a bowl mathematically (without one to measure) is difficult because they are seldom true hemispheres - they generally have flat bottoms and the slop of the sides are not uniform. You did the best thing - measure it by pouring water into it and and recording how much it held.
Generally, regardless of the shape or size, the manufacturer will state the cups/quarts that the container will hold - this is the maximum up to the brim volume. I would just go with the quantity the manufacturer states on the package - or on the item if that is where it is marked.