Cone and Cylinder

1. Content

The exhibition items showcase the volume relationship between cylinders and cones to the audience. The background board design is a question mark, symbolizing to inspire the audience to think. The question mark background board is embedded with a rotatable disk, which contains cylindrical and conical containers inside. The containers are interconnected with each other. The container is filled with green antifreeze, and the liquid can flow through each container.

When participating, the audience can rotate the disc and observe the flow of liquid between the cylindrical container and the conical container through the organic glass disc, and understand the volume relationship between the cone and the cylinder by combining the graphic or explanatory board.

Through interaction, we will find that in the case of equal base and height, the volume of a cylinder is three times that of a cone, and the volume of a cone is one-third that of a cylinder; When the volumes are equal, if the base area of the cylinder cone is equal, the height of the cone is three times that of the cylinder, and the height of the cylinder is one-third that of the cone; If the heights are equal, the base area of a cone is three times that of a cylinder, and the base area of a cylinder is one-third that of a cone. Among them, the bottom radius of the small cylinder is half of that of the large cylinder, so the volume of the small cylinder is one fourth of that of the large cylinder. The liquid inside four small cylinders can fill one large cylinder perfectly.

2. Principle

There is a fixed line and a moving line in the same plane. When the plane rotates around this fixed line once, the surface formed by this moving line is called the rotating surface, the fixed line is called the axis of the rotating surface, and the moving line is called the generatrix of the rotating surface. If the generatrix is a straight line parallel to the axis, the resulting rotating surface is called a cylindrical surface. If two planes perpendicular to the axis are used to cut a cylindrical surface, the geometric body enclosed by the two sections and the cylindrical surface is called a straight cylinder, abbreviated as a cylinder.

Cone is a geometric shape with two definitions. Definition of analytic geometry: A spatial geometric figure composed of a conical surface and a plane intersecting it (satisfying the intersection line as a circle) is called a cone. Definition of solid geometry: The geometric body enclosed by a curved surface formed by rotating the right side of a right angled triangle 360 degrees around a straight line as the axis of rotation is called a cone. The axis of rotation is called the axis of a cone. The surface formed by rotating the edge perpendicular to the axis is called the base of a cone. The surface formed by the rotation of edges that are not perpendicular to the axis is called the side of a cone. No matter what position it rotates to, the edges that are not perpendicular to the axis are called the generatrix of the cone.

In the case of equal base and height, the volume of a cylinder is three times that of a cone, and the volume of a cone is one-third that of a cylinder; When the volumes are equal, if the base area of the cylinder cone is equal, the height of the cone is three times that of the cylinder, and the height of the cylinder is one-third that of the cone; If the heights are equal, the base area of a cone is three times that of a cylinder, and the base area of a cylinder is one-third that of a cone.

3. Purpose

The exhibition items showcase the volume relationship between cylinders and cones to the audience through interactive operations, presenting abstract mathematical knowledge in a visual and intuitive form.