Introduction
Today we will learn how to find the volume of cones, cylinders, and spheres. We will also learn how their volumes are related to one another.
The Volume of a Cylinder
The top and bottom of a cylinder are two congruent circles, called bases. The height h of a cylinder is the perpendicular distance between these bases, and the radius r of a cylinder is simply the radius of the circular bases.
Even though a cylinder is technically not a prism, they share many properties. In both cases, we can find the volume by multiplying the area of their base by their height. This means that a cylinder with radius r and height h has a volume
Cones
A cone is a three-dimensional solid that has a circular base. Its side “tapers upwards” and ends in a single point called the vertex.
The radius of the cone is the radius of the circular base, and the height of the cone is the perpendicular distance from the base to the vertex.
We previously found the volume of a cylinder by approximating it using a prism. Similarly, we can find the volume of a cone by approximating it using a pyramid
Spheres
A sphere is a three-dimensional solid consisting of all points that have the same distance from a given center. This distance is called the radius of the sphere.
To find the volume of a sphere, we once again have to use Cavalieri’s Principle. Let’s start with a hemisphere – a sphere cut in half along the equator. We also need a cylinder with the same radius and height as the hemisphere, but with an inverted cone “cut out” in the middle.
Task
- How to find the volume of cones, cylinders, and spheres.
- How their volumes are related to one another.
Process
V = πr^2h. r is the radius of the circular base of the cylinder and h is the height of the cylinder (the distance between the two circular ends).
V = 1/3πr^2h. r is the radius of the circle at the base and h is the distance between the center point of the circular base and the point at the top of the cone.
V = 4/3πr^3. If I cut the sphere exactly in half I could see a circle on the face. r is the radius of the circle at the very center of the sphere.
Conclusion
The 1/3 comes from the volume formula for general cones and pyramids. The volume formula for a general cylinder is the area of the cylinder’s base times the height of the cylinder, and a cylinder can be decomposed into three cones, each with equal volume; thus, the volume of each of the cones is 1/3 the volume of a cylinder.