Volume of Composite 3D Solid Figures: Fifth Grade Mathematics

The volume of a composite three-dimensional solid is determined by decomposing the solid into non-overlapping rectangular prisms and summing their individual volumes.

The Additive Volume Principle

Volume is an additive property. If a complex solid figure can be separated into two or more distinct rectangular prisms that do not overlap, the total volume of the entire figure equals the sum of the volumes of each prism.

+--------------------------------------------------------------------------+
|                  DECOMPOSING A COMPOSITE SOLID (L-SHAPED 3D)             |
|                                                                          |
|       +----------+                                                       |
|       | Prism A  |                                                       |
|       |          +------------------+                                    |
|       |          |     Prism B      |                                    |
|       +----------+------------------+                                    |
|                                                                          |
|       Total Volume = Volume of Prism A + Volume of Prism B               |
|                    = (l_A * w_A * h_A) + (l_B * w_B * h_B)               |
+--------------------------------------------------------------------------+

Step-by-Step Decomposition Method

To find the volume of a composite 3D solid:
1. Choose whether to make a vertical cut or a horizontal cut to separate the figure into simpler rectangular prisms.

2. Label each sub-prism (Prism A, Prism B).

3. Carefully deduce any missing side lengths by examining opposite parallel edges.

4. Calculate the volume of each prism separately using V = l * w * h.

5. Add the individual volumes together.

Worked Decomposition Example:
  Given an L-shaped solid with base dimensions:
  - Total length along bottom = 10 cm.
  - Prism A stands tall on left: length = 4 cm, width = 5 cm, height = 8 cm.
  - Prism B sits low on right: height = 3 cm, width = 5 cm.
  Deduce Prism B length: Total length (10 cm) - Prism A length (4 cm) = 6 cm.
  Calculate Volume A: 4 * 5 * 8 = 160 cm^3.
  Calculate Volume B: 6 * 5 * 3 = 90 cm^3.
  Total Volume = 160 + 90 = 250 cm^3.

Checking Work with Alternative Slicing

You can verify your answer by slicing the composite solid in the opposite direction (e.g., slicing horizontally instead of vertically). Both decomposition lines must yield the exact same combined volume.

Exercises and Solutions

Exercise 1: Two-Prism Additive Decomposition. A concrete entrance step is composed of two joined rectangular prisms. The bottom landing step measures 36 inches long, 18 inches wide, and 6 inches high. The upper step attached on top measures 24 inches long, 18 inches wide, and 6 inches high. Find the total volume of concrete required to build the steps.

Solution 1: Calculate the volume of the bottom step (Prism 1): V1 = length * width * height = 36 * 18 * 6. Multiply: 36 * 18 = 648; 648 * 6 = 3,888 cubic inches. Calculate the volume of the upper step (Prism 2): V2 = 24 * 18 * 6. Multiply: 24 * 18 = 432; 432 * 6 = 2,592 cubic inches. Add the two volumes together: Total Volume = 3,888 + 2,592 = 6,480 cubic inches.

Exercise 2: Deductions of Shared Dimensions. A U-shaped wooden planter box consists of a central bottom base prism with two identical vertical side pillars attached on top. The central base has dimensions 30 cm long, 10 cm wide, and 5 cm high. Each side pillar has dimensions 8 cm long, 10 cm wide, and 15 cm high. What is the total volume of wood used in the planter box?

Solution 2: Decompose into three prisms: Base Prism and two identical Pillar Prisms. Volume of Base Prism: V_base = 30 * 10 * 5 = 1,500 cm^3. Volume of One Pillar: V_pillar = 8 * 10 * 15. Multiply: 8 * 10 = 80; 80 * 15 = 1,200 cm^3. Since there are two identical pillars: 2 * 1,200 = 2,400 cm^3. Add all three volumes: Total Volume = 1,500 + 2,400 = 3,900 cubic centimeters.