Solids of revolution: cylinder, cone and sphere
Rotating a rectangle about one side gives a cylinder: the fixed side is the axis, the opposite side is the generator, and the other two sides sweep out the two disc-shaped bases. Rotating a right triangle about one leg gives a cone: that leg is the axis, the other leg sweeps the base disc and the hypotenuse sweeps the lateral surface and is the generator. Rotating a disc about its diameter gives a ball, and its boundary, the surface made by the rotation, is the sphere. The lateral surface of a cylinder is S = 2πrh and its total surface is 2πr² + 2πrh; the lateral surface of a cone is S = πrl (r — base radius, l — generator), its total surface is πrl + πr²; the area of a sphere is S = 4πr². Unrolling the lateral surface of a cone gives a sector of radius l and arc length 2πr, so its area is (1/2) · l · 2πr = πrl.
With a teacher’s or parent’s permission: cut a sector from a paper disc (an adult handles the scissors) and roll it into a cone; check by measuring that the sector’s radius equals the cone’s generator and its arc equals the base circumference.