The volume of a rectangular prism is the number of unit cubes that fill it. Multiply length, width, and height, or multiply the base area by the height. Edges can be fractions: a prism packed with cubes of edge one half holds eight cubes in each cubic unit. Volume is written in cubic units.
Key facts:
- \(V = \ell \times w \times h\) and \(V = B \times h\) give the same volume.
- Volume is measured in cubic units such as in\(^3\) or ft\(^3\).
- A cube with edge \(\frac{1}{2}\) has volume \(\frac{1}{8}\) cubic unit.
- To find a missing height, divide the volume by the base area.
What is the volume of a rectangular prism?
Volume is the amount of space inside a solid. It is measured in cubic units. A unit cube has edges of 1 unit and a volume of 1 cubic unit. If a rectangular prism is packed with unit cubes and nothing is left over, the number of cubes is the volume.
How do you find volume when the edges are fractions?
Use smaller cubes. A cube with edges of \(\frac{1}{2}\) inch has a volume of \(\frac{1}{8}\text{ in}^3\), because 8 of them fill a 1-inch cube. A prism that is \(2\text{ in}\) by \(1\frac{1}{2}\text{ in}\) by \(1\text{ in}\) holds \(4 \times 3 \times 2 = 24\) of these cubes, so its volume is \(24 \times \frac{1}{8} = 3\text{ in}^3\). Multiplying the edges gives the same answer: \(2 \times \frac{3}{2} \times 1 = 3\).
| Formula | Meaning |
|---|---|
| \(V = \ell \times w \times h\) | length times width times height |
| \(V = B \times h\) | area of the base times height |
The two formulas agree because the base area is \(B = \ell \times w\). Think of the prism as layers: each layer holds \(B\) cubes, and there are \(h\) layers.
How do you find the volume step by step?
- Write the three edge lengths in the same unit.
- Write \(V = \ell \times w \times h\).
- Substitute the lengths. Change mixed numbers to fractions.
- Multiply, then write the answer in cubic units.
Example. A box is \(2\frac{1}{2}\text{ ft}\) long, \(1\frac{1}{2}\text{ ft}\) wide, and \(2\text{ ft}\) tall.
\(V = \frac{5}{2} \times \frac{3}{2} \times 2 = \frac{15}{4} \times 2 = \frac{15}{2} = 7\frac{1}{2}\text{ ft}^3\)
If a measurement is missing, divide: \(h = V : B\). A prism with \(V = 60\text{ ft}^3\) and \(B = 15\text{ ft}^2\) has \(h = 60 : 15 = 4\text{ ft}\).
Volume of Rectangular Prisms: frequently asked questions
What is the formula for the volume of a rectangular prism?
Volume equals length times width times height, \(V = \ell wh\). It can also be written \(V = Bh\), where \(B\) is the area of the base.
Why is volume measured in cubic units?
Volume counts how many unit cubes fill a solid. A prism that holds 30 cubes with 1-inch edges has a volume of 30 cubic inches.
How do you find volume with fractional edge lengths?
Either multiply the fractions, or pack the prism with cubes whose edge is the unit fraction. A cube with edge \(\frac{1}{3}\) has volume \(\frac{1}{27}\) cubic unit.
How is volume different from surface area?
Volume measures the space inside a solid in cubic units. Surface area measures the total area of the outside faces in square units.
How do you find a missing edge of a prism?
Divide the volume by the area of the base. For a volume of 60 cubic feet and a base area of 15 square feet, the height is \(60 : 15 = 4\) feet.
See every chapter on the Grade 6 math page.