# Volume Calculator

This volume calculator computes the volume of a cylinder, cone, sphere, prism, or pyramid. Click the desired tab to select the shape whose measurements you want to calculate.

## What is volume?

The volume of an object is the amount of three-dimensional space that the object takes up. For example, an aquarium with a length of 3 meters (m), a width of 0.5 m, and a height if 1.5 m can only hold a specific amount of water (2.25 m^{3}) before it overflows. The amount of water that the aquarium can hold is the volume (sometimes referred to as capacity) of the container.

Given that the object whose volume we want to determine is a typical shape, we can calculate its volume by plugging in its measurements into well-known formulas. Also, if the object has a shape that is made up of a few typical shapes, we can calculate the volume by finding the sum of the volumes of each shape. In cases where an object has a shape that is not typical, it is more difficult to determine its volume, and the methods for doing so will not be described on this page.

## Volume formula of typical shapes

### Cylinder

The formula for the volume of a cylinder is:

V = πr^{2}h

where π is a mathematical constant with a value of approximately 3.14, r is the radius of the base of the cylinder, and h is the height of a cylinder as measured between the circular bases of the cylinder. Thus, given a cylinder with height of 4 m and radius of 0.5 m, its volume can be calculated as follows:

V = π(0.5)^{2}(4) = π

### Cone

The formula for the volume of a cone is:

V = |
| πr^{2}h |

where π is a mathematical constant with a value of approximately 3.14, r is the radius of the base of the cone, and h is the height of the cone as measured from the center of the base to the apex of the cone. Given a cone with height 9 and radius 3, the volume of the cone can be calculated as follows:

V = |
| π(3^{2})(9) = 84.823 |

### Sphere

The formula for the volume of a sphere is:

V = |
| πr^{3} |

where π is a mathematical constant with a value of approximately 3.14 and r is the radius of the sphere. Given that the radius of a sphere is 2.5, its volume can be calculated as follows:

V = |
| π(2.5)^{3} = 65.450 |

### Prism

The formula for the volume of a prism depends on the type of prism. Squares and rectangles are prisms, and their volumes can be calculated by finding the product of their length (l), width (w), and height (h). In other words:

V = lwh

Notice that the product of the length and width of a square or rectangle is its area. The volume is simply the area of the base of the prism multiplied by its third dimension, the height. Thus, to find the volume of other types of prisms, such as a triangular prism, we just need to first find the area of the base of the prism, then multiply by the height of the prism. For example, given a triangle with a base length of 7, a height of 12, and a prism height of 14, we can calculate the volume of the prism using the formula for the area of a triangle,

A = |
| bh |

where a is area, b is the base length, and h is the height of the triangle. Thus:

A = |
| (7)(12) = 42 |

Then, multiply by the height to find the volume:

V = Ah = (42)(14) = 588

### Pyramid

The formula for the volume of a pyramid is similar to that of a prism in that it depends on the base of the pyramid. Typically, when the term "pyramid" is used, and no other type of pyramid is specified, it refers to a square pyramid. However, like prisms, pyramids can have different bases, and the formula for the volume of a pyramid is as follows:

V = |
| Ah |

where A is the area of the base, and h is the height of the pyramid as measured from the center of its base to the apex of the pyramid. In the case of a square pyramid, A = w^{2} since the length and width of the square are the same. If the base of the pyramid is some other shape, such as a triangle, we would need to first calculate the area of the triangle, then multiply by the height of the pyramid to find its volume. For example, given a pyramid with height 7 and a triangular base that has a height of 4 and a base length of 3, its area is:

A = |
| bh = |
| (3)(4) = 6 |

Thus, the volume of the pyramid is:

V = |
| Ah = |
| (6)(7) = 14 |