Lesson 6: Volume of Composite Solids

Property

The volume of a composite figure can be found by decomposing it into non-overlapping rectangular prisms and adding their individual volumes.

Examples

• An L-shaped figure can be decomposed into two rectangular prisms. The total volume is the sum of the volumes of the two smaller prisms.
• A T-shaped figure can be split into a horizontal prism and a vertical prism. Its total volume is found by adding the volumes of these two parts.
• A figure shaped like steps can be split into multiple rectangular prisms stacked on top of each other. The total volume is the sum of the volumes of all the steps.

Explanation

A composite figure is a three-dimensional shape made by combining two or more simpler shapes. To find the volume of a composite figure, we first break it down, or decompose it, into familiar shapes like rectangular prisms. After identifying the individual prisms, the next step is to find the volume of each one separately. The total volume of the composite figure is simply the sum of the volumes of all the individual prisms.

Property

The volume of a composite solid made of rectangular prisms can be found using two main methods:

• Addition: Decompose the figure into non-overlapping rectangular prisms and add their individual volumes. $V_{total} = V_1 + V_2$
• Subtraction: Enclose the figure within a larger rectangular prism and subtract the volume of the missing portion. $V_{total} = V_{large} - V_{missing}$

Examples

• Addition Method: For an L-shaped prism, you can split it into two smaller prisms. If one prism is $2 \times 4 \times 5$ and the other is $3 \times 4 \times 2$, the total volume is $(2 \times 4 \times 5) + (3 \times 4 \times 2) = 40 + 24 = 64$ cubic units.
• Subtraction Method: For the same L-shaped prism, you can imagine a large rectangular prism and subtract the empty space. If the large prism is $5 \times 4 \times 5$ and the empty space is $3 \times 4 \times 3$, the total volume is $(5 \times 4 \times 5) - (3 \times 4 \times 3) = 100 - 36 = 64$ cubic units.

Explanation

A composite prism is a 3D figure made up of two or more rectangular prisms. To find its volume, you can use the addition method by breaking the shape into smaller, familiar prisms and summing their volumes. Alternatively, the subtraction method involves calculating the volume of a larger, simpler prism that encloses the shape and then subtracting the volume of the parts that are not part of the figure. Both methods apply the standard volume formula, $V = l \times w \times h$, and will yield the same final answer.

Property

To find the volume of a composite solid made of rectangular prisms, calculate the volume of each individual prism and then add their volumes together.

Examples