Princess engagement rings 
Princess engagement rings are the second most popular shape to the round brilliant diamond.
Princess engagement rings are the ones set with square or rectangular cross cut diamonds. Diamonds with trimmed corners are called radiant. The most sought after shape is square and often is sold at a higher price than the rectangular princess with the same cut grade.
Princess cut diamonds are cheaper per carat than round diamonds of the same quality. This is dictated by the bigger demand for round stones which are easier to incorporate in designs. Although cheaper, princess engagement rings are not always better value for money. Diamonds are sold by carat which is simply a weight measurement; one carat is equal to 0.2gr. Since, to the eye, princess diamonds look smaller than round diamonds, they need more carat weight to achieve similar visual impact when set on a ring. In other words, a round diamond of 0.80ct and princess diamond of 1.1ct look similar in size, however, they are very different in weight and weight is what determines the price.
Because they are cheaper, the diamond cutters will cut a piece of rough diamond in to princess only if the will get much bigger yield. This results in limited availability in certain sizes and qualities and makes it difficult to find the optimal option when working on budget. This said, princess diamonds are beautiful stones. Princess engagement rings emanate specific style which makes them a favorite to many. My professional advice is to choose princess engagement ring only if it matches your aesthetic preferences. If it is only to save on budget then, at the end of the day, it will be not worth it.