To find the missing volume of a prism, you calculate the area of its base and multiply it by its height.
Understanding Prism Volume
The volume of any three-dimensional shape represents the amount of space it occupies. For a prism, this volume is consistently determined by the size of its base and how tall it is.
The Fundamental Formula
The core principle for calculating prism volume is simple:
Volume = Area of Base × Height
This can be written using variables as:
V = A\base × h
According to the provided information: "To calculate the volume of any prism, we find the area of the base and multiply it by the height."
Identifying the Base
A crucial first step is correctly identifying the base of the prism. The reference states: "To identify the base of the prism, look for two opposite congruent sides that are connected by any number of lateral sides." These base faces are parallel to each other. For right prisms, the lateral sides connecting the bases are always rectangles.
Steps to Calculate Prism Volume
Here is a step-by-step guide to finding the missing volume of a prism:
- Identify the Base Shape: Look at the prism and determine the shape of its congruent, parallel bases (e.g., triangle, square, rectangle, pentagon, etc.).
- Calculate the Area of the Base (A\base): Use the appropriate formula to find the area of this base shape. This is often the most variable step, depending on the type of prism.
- Identify the Height (h): The height of the prism is the perpendicular distance between the two base faces.
- Multiply Base Area by Height: Plug the calculated base area and the identified height into the formula V = A\base × h to get the volume.
Calculating the Area of the Base (A\base)
The method for finding A\base depends entirely on the shape of the prism's base. Here are formulas for common base shapes:
| Base Shape | Formula for Area (A\base) |
|---|---|
| Rectangle / Square | length × width |
| Triangle | ½ × base × height (of the triangle) |
| Circle (Cylinder) | π × radius² |
Note: A cylinder is technically a prism with a circular base.
Examples
Let's look at a couple of examples:
-
Example 1: Rectangular Prism
- Problem: A rectangular prism has a base measuring 5 cm by 4 cm, and the height of the prism is 10 cm.
- Solution:
- Base shape is a rectangle.
- Area of Base (A\base) = length × width = 5 cm × 4 cm = 20 cm².
- Height (h) = 10 cm.
- Volume (V) = A\base × h = 20 cm² × 10 cm = 200 cm³.
- The missing volume is 200 cubic centimeters.
-
Example 2: Triangular Prism
- Problem: A triangular prism has a base that is a triangle with a base length of 8 inches and a height of 6 inches. The height of the prism is 12 inches.
- Solution:
- Base shape is a triangle.
- Area of Base (A\base) = ½ × base (of triangle) × height (of triangle) = ½ × 8 inches × 6 inches = 24 inches².
- Height (h) = 12 inches.
- Volume (V) = A\base × h = 24 inches² × 12 inches = 288 inches³.
- The missing volume is 288 cubic inches.
Practical Considerations
- Always ensure that the units of your base measurements and height are consistent (e.g., all in centimeters, all in inches) before calculating the volume. The resulting volume will be in cubic units (cm³, in³, m³, etc.).
- For complex base shapes, break down the base area calculation into simpler geometric figures if necessary.
By finding the area of the base and multiplying it by the height, you can successfully determine the volume of any prism.
