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While cut refers more to quality and level of brilliance, the shape is one aspect of the cut. Review the possible shapes available on the market, and choose the one you like best. If buying a diamond for someone else, ask his/her opinion before selecting a shape, or bring a trusted friend/family member.  Select a popular, classic shape if you have no one to ask. The most popular shapes are round brilliant, princess, and cushion cut.  Consider a non-traditional shape if the receiver has a non-traditional personality. Also, note that less common shapes can be 30%, for the same karat weight and quality! Less common shapes that are still fairly popular include marquise, oval, pear, and heart. A diamond's cut determines the vast majority of its brilliance, and many consider the cut to be the most important factor when choosing a diamond. Ask the jeweler to see the Gemological Institute of America's (GIA’s) diamond grading report for any diamond you consider purchasing. This report will give you a rough grade on cut:  Select an "Ideal Cut" diamond for the highest level of perfection. Choose a "Very Good" or "Excellent" grade diamond for a slightly less expensive yet high-quality stone. Consider a "Good" grade if you want to balance the other C's and still stick to a budget. Never purchase a fair to poor quality diamond, especially for engagement rings and other gifts. Diamonds at these levels severely lack brilliance. The pavilion side is the lower side of a diamond, which isn’t visible enough and has no impact on its sparkling style.  Chevrons are of particular importance when considering the princess cut diamonds. These are the facets at the pavilion side of the diamond surrounding the main pavilion. Most princess cuts have two, three or four chevrons. The more the chevrons, the more the light gets reflected upon. There is no "right" number of chevrons, just personal preference. More chevrons lead to a more multifaceted look, whereas less has a smoother look.
Pick out the right shape for you and your budget. Choose the highest quality of cut you can afford. Consider the number of chevrons.