Volume of Rectangular Prisms - Sixth Grade Mathematics
Volume is the measure of three-dimensional interior space enclosed within a solid boundary, quantified by the number of unit cubes required to fill it completely without gaps.
The Concept of Unit Cubes and Packing
While perimeter measures 1D distance and area measures 2D coverage, volume measures 3D capacity. The standard unit of volume is the cubic unit (such as cubic centimeters cm^3, cubic inches in.^3, or cubic meters m^3).
Visualizing Packing Unit Cubes into a Prism:
Prism: Length = 4 units, Width = 3 units, Height = 2 units
Layer 1 (Bottom Layer):
A flat grid of 4 columns and 3 rows = 4 * 3 = 12 unit cubes.
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Layer 2 (Top Layer):
Stack a second identical layer of 12 unit cubes on top!
Total Volume = 12 cubes/layer * 2 layers = 24 cubic units.
The Fundamental Volume Formulas
Because each horizontal layer is the area of the base, volume is universally computed as base area multiplied by height.
Formulas for Rectangular Prisms:
Volume = length * width * height (V = l * w * h)
or
Volume = (Area of Base) * height (V = B * h, where B = l * w)
For a Cube with edge s:
Volume = s * s * s = s^3
Example 1: Rectangular shipping box with l = 12 cm, w = 5 cm, h = 6 cm
V = l * w * h = 12 * 5 * 6 = 60 * 6 = 360 cm^3.
Example 2: A cube with side length 4 inches
V = s^3 = 4^3 = 4 * 4 * 4 = 64 in.^3.
Finding Missing Dimensions Using Volume
When the total volume and two dimensions are provided, algebra allows us to isolate and solve for the unknown third dimension.
Problem: A rectangular aquarium has a volume of 3,600 cubic inches.
Its length is 24 inches and width is 10 inches. What is the height h of the aquarium?
Step 1: Write formula:
V = l * w * h
Step 2: Substitute known numbers:
3600 = 24 * 10 * h
3600 = 240 * h
Step 3: Solve for h:
h = 3600 / 240
h = 15 inches.
Check: 24 * 10 * 15 = 240 * 15 = 3,600 in.^3. (Confirmed!)
Metric Liquid Volume Equivalences
In the metric system, solid volume and liquid capacity are directly linked: 1 cubic centimeter (1 cm^3 or 1 cc) = 1 milliliter (1 mL). 1,000 cubic centimeters (1,000 cm^3) = 1 liter (1 L). Therefore, a rectangular container measuring 10 cm by 10 cm by 10 cm has a volume of 1,000 cm^3 and holds exactly 1 liter of liquid.
Exercises
Exercise 1. Calculate the volume of each rectangular prism: (a) Length 15 cm, width 8 cm, height 5 cm; (b) A cube with edge length 9 inches; (c) Base area B = 45 square meters, height = 7 meters; (d) Length 2.5 meters, width 1.2 meters, height 3 meters.
Exercise 2. A rectangular shipping container has a volume of 840 cubic feet. The length is 14 feet and the width is 10 feet. Find the height of the container.
Exercise 3. A rectangular prism has a square base with sides of 6 cm. If the volume is 288 cm^3, what is the height of the prism?
Exercise 4. A swimming pool is 25 meters long, 10 meters wide, and 2 meters deep throughout. (a) What is the volume of the pool in cubic meters? (b) How many liters of water does the pool hold when completely full? (1 m^3 = 1,000 liters).
Solutions
Solution to Exercise 1. For (a), V = l * w * h = 15 * 8 * 5 = 120 * 5 = 600 cm^3. For (b), V = s^3 = 9^3 = 9 * 9 * 9 = 729 in.^3. For (c), V = B * h = 45 * 7 = 315 m^3. For (d), V = 2.5 * 1.2 * 3 = 3 * 3 = 9 m^3.
Solution to Exercise 2. Using V = l * w * h: 840 = 14 * 10 * h. 840 = 140 * h. Divide by 140: h = 840 / 140 = 6 feet. The container is 6 feet high.
Solution to Exercise 3. The square base has area B = 6 * 6 = 36 cm^2. Using V = B * h: 288 = 36 * h. Divide by 36: h = 288 / 36 = 8 cm. The height of the prism is 8 cm.
Solution to Exercise 4. For (a), Volume = 25 * 10 * 2 = 500 m^3. For (b), since 1 m^3 = 1,000 liters: Total capacity = 500 * 1,000 = 500,000 liters of water.