A couple of unexpected visitors change your dinner to a party of 10 instead of 8 guests, requiring you to create two more slices than you had intended. It is easy to cut a pie into 8 slices by simply cutting it into quarters and cutting each quarter in half. If you think of the pie not as a circle, but as a pentagon shape with 5 points, it's much easier to cut 10 slices. Toothpicks work well as visual aids until you are able to visualize the cuts yourself.
Visualize an equilateral pentagon shape inside the larger circle of pie with each point of the pentagon occupying a point along the edge of the crust. To make it even easier, cut out an equilateral pentagon from card stock and lay it on top of the pie to use as a guide.
Push a toothpick into the edge of the pie at each of the 5 pentagon points. Push the toothpick into the pie crust, or slip it between the pie crust and pie pan. You can push the toothpicks into the crust itself without leaving any noticeable holes by making your cuts over the holes.
Insert another 5 toothpicks into the pie at the center point between each of the 5 original toothpicks, dividing the pie into equal 10ths.
Insert an 11th toothpick directly in the center of the pie. This helps make the cuts straight and the slices perfectly even. You can skip the toothpick and instead mentally visualize the center point.
Remove 1 toothpick from the pie. Line up your knife with the hole where you removed the toothpick. Drag the knife straight across the pie to the opposite toothpick, removing and replacing the toothpick at the center of the pie as you pass through center. Remove the toothpick opposite the first 1, and finish the cut through the small toothpick hole. You can also make the cuts directly beside the toothpicks, but removing them first eliminates any visible holes.
Start the second cut at the toothpick to the right of the first, crossing to the toothpick to the left of the end of the first cut. Repeat this step with the remaining toothpicks, starting the third cut at the toothpick to the left of the first toothpick, crossing to the right of the end of the first cut. It takes a total of 5 cuts, including the first cut for the first pair of toothpicks.
The toothpicks are not necessary for cutting a pie into 10 slices. If you feel comfortable visualizing each cut without the toothpicks or pentagon template, then you can cut it without any guide. Whether you use toothpicks or not, the process requires 5 total cuts: 1 cut to divide the pie in half, followed by 4 equidistant cuts across the pie.
Each wedge of the pie represents a 36-degree angle. A circle is 360 degrees, which, divided by 10 slices, equals 36 degrees per slice.