Volume of Cone

The volume of a cone is the measure of space inside it, equal to one-third the volume of a cylinder that has the exact same base radius and height.

Comparing a cylinder and a cone

When a cylinder and a cone share the same vertical height and the same circular base radius, the cylinder has three times more volume than the cone. If a cone is filled with peanuts and emptied into the cylinder, it must be filled three times to fill the cylinder to the top. This means that two full cones hold less than one cylinder of the same size, which explains why two cone-shaped snacks empty much faster than one cylinder-shaped container.

Formulas for cylinder and cone

• The base of a cylinder is a flat circle, and its area is calculated using the formula pi r squared, where r represents the radius.
• Adding the dimension of height, represented by h, gives the volume of a cylinder as pi r squared multiplied by h.
• Because three identical cones fill one cylinder, dividing the volume of the cylinder by three gives the volume of one cone.
• This calculation establishes the volume formula of a cone as pi r squared h divided by three.

Measuring hemispheres

A solid round shape called a sphere can be cut into two equal halves, with each half known as a hemisphere. To measure its capacity, take a cone that has the exact same radius and height as the hemisphere. Filling this cone with grains or peanuts twice and pouring them into the hemisphere fills it completely. As a result, the volume of one hemisphere equals the volume of two identical cones.

Volume of a full sphere

• A complete sphere consists of two hemispheres joined together.
• Because each hemisphere holds the volume of two cones, a whole sphere holds the volume of four identical cones.
• Multiplying the volume of a cone by four yields the volume formula for a sphere as four-thirds pi r squared h.
• Comparing these geometric shapes shows that a ball-shaped container holds considerably more than a cone-shaped container of the same dimensions.