Volume of Rectangular Prisms (l x w x h and B x h): Fifth Grade Mathematics

The volume of a right rectangular prism is calculated using two equivalent formulas: V = length * width * height and V = Base Area * height.

The Two Volume Formulas

Because counting individual unit cubes becomes impractical for large objects, fifth-grade mathematicians use algebraic formulas to calculate volume.

+--------------------------------------------------------------------------+
|                  THE TWO RECTANGULAR PRISM VOLUME FORMULAS               |
|                                                                          |
|   Formula 1:  V = l * w * h                                              |
|               (Volume = length * width * height)                         |
|                                                                          |
|   Formula 2:  V = B * h                                                  |
|               (Volume = Base Area * height)                              |
|               Where capital B represents the 2D area of the base (l * w).|
+--------------------------------------------------------------------------+

Connecting the Two Formulas:
      B = l * w
      V = (l * w) * h  --->  V = B * h
  Both formulas calculate the exact same physical volume!

Step-by-Step Volume Calculation

To calculate the volume of a rectangular prism:
1. Identify the three dimensions: length, width, and height. Ensure all three measurements share identical units!

2. Multiply the dimensions together. By the Associative Property of Multiplication, you can multiply the three numbers in any order.

3. Label the answer with cubic units (e.g., cm^3, in^3, ft^3).

Calculating Volume of Prism (8 cm by 5 cm by 6 cm):
  V = 8 * 5 * 6
  Step 1: Base Area B = 8 * 5 = 40 sq cm.
  Step 2: Multiply by height: 40 * 6 = 240 cu cm.
  Volume = 240 cm^3.

Volume of a Cube

A cube is a special rectangular prism where length, width, and height are all equal (l = w = h = s). Formula: V = s * s * s = s^3. For a cube with side 5 inches: V = 5^3 = 5 * 5 * 5 = 125 cubic inches.

Exercises and Solutions

Exercise 1: Applying Both Volume Formulas. A rectangular shipping crate measures 12 feet long, 6 feet wide, and 8 feet tall. Calculate the volume using Formula 1 (l * w * h). Then identify the Base Area B and calculate the volume using Formula 2 (B * h).

Solution 1: Formula 1: V = length * width * height = 12 * 6 * 8. Multiply 12 * 6 = 72; then 72 * 8 = 576 cubic feet. Formula 2: The base is a rectangle measuring 12 feet by 6 feet, so Base Area B = 12 * 6 = 72 square feet. Multiply Base Area by height: V = B * h = 72 * 8 = 576 cubic feet. Both formulas confirm the volume is 576 cubic feet (ft^3).

Exercise 2: Finding a Missing Dimension from Volume. A rectangular aquarium holds a volume of 4,320 cubic inches of water. The aquarium has a base length of 24 inches and a base width of 15 inches. What is the height of the aquarium?

Solution 2: Step 1: Find the base area B of the aquarium: B = length * width = 24 * 15 = 360 square inches. Step 2: Use the formula V = B * h to solve for the missing height: 4,320 = 360 * h. Divide the total volume by the base area: h = 4,320 / 360 = 12 inches. The height of the aquarium is 12 inches.