As shown in the diagram below, a triangular prism can be thought of as half of a rectangular prism. Thus, its volume can be calculated as follows:

Generally, any prism can be divided into multiple triangular prisms. Therefore, the volume of a prism is equal to the sum of the volumes of the triangular prisms it contains. As a result, the volume of a prism can be expressed as:

For a cylinder, which can be considered a prism with circular bases:

If water is poured from a triangular pyramid-shaped container into a prism-shaped container with the same base area and height, the water will fill one-third of the prism. Thus, the volume of a pyramid is of the volume of a prism with the same base and height.

Similarly, the volume of a cone is of the volume of a cylinder with the same base and height.

For a cone with base radius and height :

When a hemisphere-shaped container is filled three times with sand, it completely fills a cylinder-shaped container with the same base radius and height. Hence, the volume of a hemisphere is of the volume of the cylinder.

Thus, the volume of a sphere, which is twice the volume of a hemisphere, is:

Where is the radius of the sphere.