Composite 3D figures (surface area and volume)
Composite Three-Dimensional Figures
Composite Three-Dimensional Figures is a topic in Three-Dimensional Measurement in the California Common Core State Standards. It is aligned to Standard G-GMD.3, which requires students to use volume and surface-area formulas for composite three-dimensional figures to solve problems.
A composite solid is split into familiar solids whose volumes and surface areas are added or subtracted.
Theory
A composite solid is built from several simple solids. Find its volume by decomposing:
- Add the volumes of joined solids (a cylinder plus a hemisphere).
- Subtract the volume of a hollow part or drilled hole.
The strategy:
How to find a composite volume
- Break the solid into familiar solids.
- Compute each volume with its formula.
- Add joined parts; subtract hollow parts.
- Keep \(\pi\) exact when round solids appear.
Add the cylinder and hemisphere volumes.
| \(\text{cylinder}\) | \(=\) | \(\pi(3)^2(8)=72\pi\) |
| \(\text{hemisphere}\) | \(=\) | \(\dfrac12\cdot\dfrac43\pi(3)^3=18\pi\) |
| \(\text{total}\) | \(=\) | \(72\pi+18\pi=90\pi\) |
Subtract the cylinder from the cube.
| \(\text{cube}\) | \(=\) | \(6^3=216\) |
| \(\text{cylinder}\) | \(=\) | \(\pi(1)^2(6)=6\pi\) |
| \(\text{remaining}\) | \(=\) | \(216-6\pi\) |
Add the two volumes.
| \(\text{cylinder}\) | \(=\) | \(\pi(4)(5)=20\pi\) |
| \(\text{cone}\) | \(=\) | \(\dfrac13\pi(4)(3)=4\pi\) |
| \(\text{total}\) | \(=\) | \(24\pi\) |
Break it into familiar solids, find each volume, and add joined parts or subtract hollow parts.
Common pitfalls
Frequently asked questions
How do you find the volume of a composite solid?
Split it into familiar solids, find each volume, and add joined parts or subtract hollow ones.
What is the volume of a hemisphere?
Half a sphere: \(\dfrac12\cdot\dfrac43\pi r^3=\dfrac23\pi r^3\).
When do you subtract volumes?
When part of the solid is hollow or drilled out — subtract that volume from the whole.
How is this like composite 2D area?
The same decompose-and-combine idea, but with volumes of solids instead of areas of shapes.