The exact formula for the volume of a parallelogram prism is V = A_base × h_prism, where A_base is the area of the parallelogram base and h_prism is the perpendicular height of the prism.
What is the Formula for the Volume of a Parallelogram Prism?
The volume of any prism, including a parallelogram prism, is fundamentally calculated by multiplying the area of its base by its perpendicular height. For a parallelogram prism, the base is a parallelogram, which requires a specific approach to calculate its area.
Understanding the General Volume Formula for Prisms
The general formula for the volume of any prism is:
$$V = A{\text{base}} \times h{\text{prism}}$$
Where:
- V represents the volume of the prism.
- A_base represents the area of the prism's base.
- h_prism represents the perpendicular height of the prism (the distance between its two parallel bases).
This principle applies universally across all types of prisms, whether their bases are triangles, rectangles, or, in this case, parallelograms. To delve deeper into prism volumes, you can explore resources on the volume of a prism.
Calculating the Area of a Parallelogram Base
Since the base of a parallelogram prism is a parallelogram, its area must be calculated first. The formula for the area of a parallelogram is:
$$A{\text{parallelogram}} = b{\text{base}} \times h_{\text{parallelogram}}$$
Where:
- A_parallelogram is the area of the parallelogram.
- b_base is the length of one side of the parallelogram (acting as its base).
- h_parallelogram is the perpendicular height of the parallelogram relative to the chosen base (the shortest distance between the chosen base and the opposite side).
It's important to distinguish between h_parallelogram (the height of the base) and h_prism (the height of the entire prism). For more information on finding the area of a parallelogram, refer to guides on the area of a parallelogram.
The Specific Formula for a Parallelogram Prism
Combining the general prism volume formula with the specific base area for a parallelogram, we get the detailed formula for the volume of a parallelogram prism:
$$V = (b{\text{base}} \times h{\text{parallelogram}}) \times h_{\text{prism}}$$
This formula can be understood as:
| Variable | Description |
|---|---|
| V | The total volume of the parallelogram prism. |
| b_base | The length of the base side of the parallelogram that forms the prism's base. |
| h_parallelogram | The perpendicular height of the parallelogram base (measured from b_base to the opposite side). |
| h_prism | The perpendicular height of the entire prism (the distance between the two parallelogram bases). |
Practical Example and Solution
Let's consider a parallelogram prism with the following dimensions:
- Length of the base side of the parallelogram (
b_base): 10 cm - Height of the parallelogram base (
h_parallelogram): 5 cm - Height of the prism (
h_prism): 12 cm
To find its volume:
-
Calculate the area of the parallelogram base:
- A_base =
b_base×h_parallelogram - A_base = 10 cm × 5 cm
- A_base = 50 cm²
- A_base =
-
Calculate the volume of the prism:
- V = A_base ×
h_prism - V = 50 cm² × 12 cm
- V = 600 cm³
- V = A_base ×
Therefore, the volume of the parallelogram prism is 600 cubic centimeters.
