Composite 3D figures (surface area and volume)
Composite Three-Dimensional Figures
Composite Three-Dimensional Figures is a topic in Three-Dimensional Measurement in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.11(C), G.11(D), which requires students to apply surface area and volume formulas to composite three-dimensional figures.
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.