Section 14.4: Volumes of Prisms

Property

Evaluate expressions at specific values of their variables. Include expressions that arise from volume formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas $V=s^3$ for a cube and $V=lwh$ for a rectangular prism to find volumes.

Examples

• Find the volume of a cube with side length $s = 4$ cm using the formula $V=s^3$. Substitute $s=4$ to get $V = 4^3 = 4 \cdot 4 \cdot 4 = 64$ cubic cm.
• Find the volume of a rectangular prism with length $l = 6$ ft, width $w = 3$ ft, and height $h = 2$ ft using the formula $V=lwh$. Substitute the values to get $V = 6 \cdot 3 \cdot 2 = 36$ cubic ft.
• The volume of a cube is $V=s^3$. If the side length $s$ is $2.5$ meters, the volume is $V = (2.5)^3 = 2.5 \cdot 2.5 \cdot 2.5 = 15.625$ cubic meters.

Explanation

To evaluate a volume expression, you replace each variable with its given number and solve using order of operations. It's like using a recipe: the volume formula is your instructions, the variable values are your ingredients, and the final answer is your finished calculation. Always remember that volume is measured in cubic units.

Property

The volume of a prism is the product of the height by the area of the base.
That is, if the area of the base is $B$ and the height is $h$, volume is $V = Bh$.

Examples

• A rectangular prism with a base of $5 \text{ cm} \times 6 \text{ cm}$ and a height of $12 \text{ cm}$ has a volume of $V = (5 \times 6) \times 12 = 360 \text{ cm}^3$.

• A triangular prism has a base area of $30 \text{ in}^2$ and a height of $7 \text{ in}$. Its volume is $V = 30 \times 7 = 210 \text{ in}^3$.

Property

The volume of any prism can be calculated using the formula $V = B \times h$, where $B$ is the area of the base and $h$ is the height. Different prisms with the same base area and height will have the same volume, regardless of the shape of their base.

Examples