Volume
Volume questions are formula plug-ins. Memorize the six SAT volume formulas and these become 30-second wins — the trick is keeping units straight.
Volume measures how much 3D space a solid occupies. Units are cubed: cubic feet, cubic centimeters, etc.
The six SAT volume formulas:
| Solid | Volume | Surface area |
|---|---|---|
| Cube (side $s$) | $s^3$ | $6s^2$ |
| Box (l × w × h) | $lwh$ | $2(lw + lh + wh)$ |
| Cylinder | $\pi r^2 h$ | $2\pi r^2 + 2\pi rh$ |
| Sphere | $\frac{4}{3} \pi r^3$ | $4\pi r^2$ |
| Cone | $\frac{1}{3} \pi r^2 h$ | $\pi r^2 + \pi r \ell$ (slant $\ell$) |
| Pyramid (square base) | $\frac{1}{3} s^2 h$ | varies |
1. Rectangular prism (box): — length × width × height.
2. Cube: — side cubed (special case of a box where ).
3. Cylinder: — base area × height. The base is a circle.
4. Sphere: .
5. Cone: — one-third of the cylinder with the same base and height.
6. Pyramid: . For a square pyramid, base area = , so .
The pattern: the cone and pyramid formulas both have a — they're tapered versions of the cylinder and prism.
| Tapered solid | Volume | Equivalent unfolded solid | Volume ratio |
|---|---|---|---|
| Cone | $\frac{1}{3} \pi r^2 h$ | Cylinder ($\pi r^2 h$) | 1 : 3 (cone is 1/3 of cylinder) |
| Pyramid | $\frac{1}{3} bh$ | Prism ($bh$) | 1 : 3 (pyramid is 1/3 of prism) |
Surface area is sometimes asked too:
- Cube SA = .
- Box SA = .
- Sphere SA = .
- Cylinder SA = (two circular bases + curved side).
SAT-typical setups:
- "A box has dimensions..." → plug into .
- "A cylinder of radius and height ..." → .
- "How does the volume change if you double the radius?" → tests knowledge that volume scales with (or for cylinders depending on what changes).
- "How much water can fill..." → just compute volume; convert units if needed.
Scaling rule (similar 3D figures): if linear dimensions scale by , volume scales by .
Example: doubling all dimensions of a box multiplies volume by .
Unit traps: a volume of 60 cubic inches converted to cubic feet is NOT 60/12. It's cubic feet. Cubic conversions cube the linear factor.
Reference sheet: the SAT provides the formulas for sphere, cone, pyramid at the start of the Math section — but knowing them by heart is faster.
Identify the solid (box / cube / cylinder / sphere / cone / pyramid) and plug into the formula. Watch for diameter-vs-radius and unit traps.
A cylinder has radius 6 and height 9. What is its volume? (Express in terms of π)
Worked examples
A cylindrical water tank has radius 3 feet and height 8 feet. What is its volume in cubic feet?
A solid cone has radius 6 and height 9. A sphere has radius 6. Which solid has the larger volume, and by how much?
Common pitfalls
Cone: . Pyramid: . Without the , you compute the volume of the cylinder/prism that contains the cone or pyramid — three times too big.
Cylinder: (radius squared, height to the first power). Sphere: (radius cubed). Mixing these up gives an answer off by a factor of .
ft in, so ft³ in³. NOT 12 in³. To convert volume between units, cube the linear factor.
Volume formulas use radius. If the problem gives a diameter, divide by 2 first. Sphere with diameter 8 has radius 4, not 8.
Key takeaways
Box: . Cube: .
Cylinder: . Sphere: .
Cone: . Pyramid: .
Cone/pyramid = one-third of the cylinder/prism with the same base and height.
Doubling all linear dimensions multiplies volume by . Cubic conversions cube the factor.
Watch & learn
Curated Khan Academy walkthroughs on Volume. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.
Try it yourself
5 practice questions on Volume, drawn from the question bank. The tutor is one click away if you get stuck.