Decide on the number of sides your polygon will have. This can be any number above two, but numbers above ten will be difficult to construct. For example, you could make a hexagon.
Place the appropriate number of toothpicks on the table. For a hexagon, place six toothpicks.
Divide 360 by the number of sides. For the hexagon, 360/6 = 60.
Place two toothpicks with an angle of that many degrees between them. Use the protractor to measure the angle. You could keep the toothpicks in place by taping them down.
Repeat this until all the toothpicks are gone.
Glue the toothpicks together in this position.
Decide on the number of sides your polygon will have. For example, you could make a triangle.
Place more toothpicks than than the sides of the polygon on the table. For example, you could make a triangle with sides that are two, two and one toothpick long. You would then need five toothpicks.
Glue some toothpicks together in a straight line. The line can consist of any number of toothpicks, but if you use more than about three in one line, it may be hard to construct the polygon. For the example, glue two pairs of toothpicks together. Let the glue set before continuing.
Repeat step 3 if you wish. That is, one or more sides can have length greater than one. In the example you make a triangle that has two sides of 2 toothpicks in length.
Lay the toothpicks and connected toothpicks out so they have the desired number of sides and glue them together.