Geometry Reference
3D Measurement: Volume and Surface Area
Volume measures the space inside a solid; surface area measures the material needed to cover its outside.
Solved Examples
Jump directly to a worked example by solid and measurement type.
Fact Table
| Solid | Volume | Surface Area |
|---|---|---|
| Prism | $V=Bh$ | sum of every face |
| Cylinder | $V=\pi r^2h$ | $SA=2\pi r^2+2\pi rh$ |
| Cone | $V=\frac13\pi r^2h$ | $SA=\pi r^2+\pi r\ell$ |
| Sphere | $V=\frac43\pi r^3$ | $SA=4\pi r^2$ |
Content Formulas
Rectangular Prism
$$V=\ell wh,\qquad SA=2(\ell w+\ell h+wh)$$
Slant Height
$$\ell=\sqrt{r^2+h^2}$$
Units
$$\text{area: square units}\qquad\text{volume: cubic units}$$
Use slant height for a cone’s lateral surface area, but use vertical height for its volume.
Classic Examples
Measure a Rectangular Prism
A rectangular prism has length $5$, width $3$, and height $4$. Find its volume and surface area.
Solution Steps
- Multiply all three dimensions for volume.
- Use the three face-pair products for surface area.
- State both measurements with units.
$$V=5(3)(4)=60$$$$SA=2(5\cdot3+5\cdot4+3\cdot4)=2(47)$$$$\boxed{V=60\text{ cubic units},\ SA=94\text{ square units}}$$
Find a Cylinder’s Volume
Find the volume of a cylinder with radius $3$ and height $4$.
Solution Steps
- Use the cylinder volume formula.
- Square the radius before multiplying by the height.
- State the exact volume.
$$V=\pi r^2h$$$$V=\pi(3^2)(4)=36\pi$$$$\boxed{36\pi\text{ cubic units}}$$
Find a Cone’s Surface Area
A cone has radius $3$ and vertical height $4$. Find its total surface area.
Solution Steps
- Use the $3$-$4$-$5$ triangle to find slant height.
- Add the base area and lateral area.
- State the exact total surface area.
$$\ell=\sqrt{3^2+4^2}=5$$$$SA=\pi(3^2)+\pi(3)(5)$$$$\boxed{SA=24\pi\text{ square units}}$$
Find a Sphere’s Volume
Find the volume of a sphere with radius $3$.
Solution Steps
- Use the sphere volume formula.
- Cube the radius.
- Simplify the exact result.
$$V=\frac43\pi r^3$$$$V=\frac43\pi(3^3)=\frac43\pi(27)$$$$\boxed{V=36\pi\text{ cubic units}}$$