Since the cube, prism, and pyramid are three of the basic solid objects in geometry, it is important to know how to determine the volume of the cube, prism, and pyramid. In mathematics and the natural and engineering sciences, the properties of these objects are of great importance. Most of the time, the geometric and physical properties of a more complex object are always approximated using the properties of the solid objects. Volume is one such property.

## How to find the volume of a cube

Cube is a solid object with six square faces that meet at right angles. It has 8 vertices and 12 edges, and its edges are of equal length. The volume of the cube is the fundamental (perhaps the easiest volume to determine) of the volume of all solid objects. The volume of a cube is given by

**V _{cube} = a ^{3}** , where

*a is*the length of its edges.

## How to find the volume of a prism

A prism is a polyhedron; it is a solid object that consists of two congruent (similar shape and size) polygonal surfaces whose identical edges are connected by rectangles. The polygonal area is called the base of the prism, and the two bases are parallel to each other. However, it is not necessary that they are positioned exactly one above the other. If they are positioned exactly on top of each other, the rectangular sides and the base meet at right angles. This type of prism is called a right angle prism.

If the base area (polygon area) is A and the vertical height between the base areas is h, then the volume of a prism is given by the formula:

**V- _{prism} = Ah**

The result applies regardless of whether it is a right-angled prism or not.

## How to find the volume of a pyramid

The pyramid is also a polyhedron with a polygonal base and a point (called the apex) connected by triangles extending from the edges. A pyramid has only one vertex, but the number of vertices depends on the polygonal base.

The volume of a pyramid with a base area A and a height perpendicular to the vertex h is given by

**V- _{pyramid} = 1/3 Ah**

## How to Find the Volume of a Cube, Prism, and Pyramid Method

### Volume of a cube

The cube is the easiest solid object to find volume.

- Find the length of a side (consider a)
- Increase this value by 3, i.e. a
^{3}(find the die) - The resulting value is the volume of the cube.

The unit of volume is the cubic meter of the unit of measure in which length was measured. So if the sides were measured in meters, then the volume is given in cubic meters.

### Volume of a prism

- Determine the area of one of the bases of the prism (A) and determine the vertical height between the two bases (h).
- The product of the area h and the vertical height gives the volume of the prism.

Note: This result applies to every type of prism, regardless of whether it is regular or non-regular.

### Volume of a pyramid

- Find the area of the base of the pyramid (A) and determine the vertical height from the base to the top (h).
- Remove the product from the base and the vertical height. One third of the resulting values is the volume of the pyramid.

Note: This result applies to every type of prism, regardless of whether it is regular or non-regular.

## How to find the volume of a cube, prism and pyramid - examples

### Find the volume of a cube

1. One edge of the cube is 1.5 m long, find the volume of the cube.

- The length of the cube is given as 1.5 m. If not stated directly, determine the length by other geometric means or by measuring.
- Take the cube of length. That is (1.5)
^{3}= 1.5 × 1.5 × 1.5 = 3.375m^{3} - One cube has a volume of 3.375 cubic meters.

### Find the volume of a prism

2. A triangular prism has a length of 20cm. The base of the prism is an isosceles triangle with equal sides ^{forming} an angle of 60 ^{°} . If the length of the side opposite the angle is 4 cm, find out the volume of the pyramid.

- First, determine the area of the base. Using trigonometric ratios, we can determine the vertical height of the base triangle from the 4cm edge to the opposite vertex as 2 tan 60
^{0}= 2 × √3≅3.4641 cm. Therefore, the area of the base is 1/2 × 4 × 3.4641 = 6.9298 cm^{2} - The vertical height is given (as length) as 20cm. Now we can calculate the volume by multiplying the area of the base by the vertical height, e.g. B. V-
_{prism}= A x h = 6.9298 cm^{2}x 20 cm = 138.596 cm^{3}. - The volume of the pyramid is 138.596 cm
^{3}.

### Find the volume of a pyramid

3. A rectangular right pyramid has a base area of 40 m wide and 60 m long. If the height to the top of the pyramid is 20 m from the base, find the volume enclosed by the surface of the pyramid.

- The base area can easily be determined by the product of the lengths of the two sides. Therefore the area is 40m × 60m = 2400m
^{2} - The vertical height is given as 20m. Therefore the volume of the pyramid V
_{pyramid}= 1/3 × 2400m^{2}× 20m = 16,000m^{3}