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Math

Volume

4 min readEasy10-question checkpoint

Volume questions are formula plug-ins. Memorize the six SAT volume formulas and these become 30-second wins — the trick is keeping units straight.

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Volume measures how much 3D space a solid occupies. Units are cubed: cubic feet, cubic centimeters, etc.

The six SAT volume formulas:

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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): $V = lwh$ — length × width × height.

2. Cube: $V = s^3$ — side cubed (special case of a box where $l = w = h = s$ ).

3. Cylinder: $V = \pi r^2 h$ — base area × height. The base is a circle.

4. Sphere: $V = \frac{4}{3} \pi r^3$ .

5. Cone: $V = \frac{1}{3} \pi r^2 h$ — one-third of the cylinder with the same base and height.

6. Pyramid: $V = \frac{1}{3} \cdot \text{(base area)} \cdot h$ . For a square pyramid, base area = $s^2$ , so $V = \frac{1}{3} s^2 h$ .

The pattern: the cone and pyramid formulas both have a $\frac{1}{3}$ — they're tapered versions of the cylinder and prism.

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Cone vs cylinder, pyramid vs prism — the $\frac{1}{3}$ pattern

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 = $6s^2$ .
Box SA = $2(lw + lh + wh)$ .
Sphere SA = $4\pi r^2$ .
Cylinder SA = $2\pi r^2 + 2\pi rh$ (two circular bases + curved side).

SAT-typical setups:

"A box has dimensions..." → plug into $V = lwh$ .
"A cylinder of radius $r$ and height $h$ ..." → $\pi r^2 h$ .
"How does the volume change if you double the radius?" → tests knowledge that volume scales with $r^3$ (or $r^2$ 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 $k$ , volume scales by $k^3$ .

Example: doubling all dimensions of a box multiplies volume by $2^3 = 8$ .

Unit traps: a volume of 60 cubic inches converted to cubic feet is NOT 60/12. It's $60 / (12)^3 = 60/1728 \approx 0.035$ 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.

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Quick check

Identify the solid (box / cube / cylinder / sphere / cone / pyramid) and plug into the formula. Watch for diameter-vs-radius and unit traps.

A rectangle has a length of 14 cm and a width of 9 cm. What is the area of the rectangle, in square centimeters?

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Worked examples

Example 1

A cylindrical water tank has radius 3 feet and height 8 feet. What is its volume in cubic feet?

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Example 2

A solid cone has radius 6 and height 9. A sphere has radius 6. Which solid has the larger volume, and by how much?

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Common pitfalls

Forgetting the $\frac{1}{3}$ on cone or pyramid

Cone: $V = \frac{1}{3} \pi r^2 h$ . Pyramid: $V = \frac{1}{3} \cdot \text{base} \cdot h$ . Without the $\frac{1}{3}$ , you compute the volume of the cylinder/prism that contains the cone or pyramid — three times too big.

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Squaring vs cubing the wrong factor

Cylinder: $\pi r^2 h$ (radius squared, height to the first power). Sphere: $\frac{4}{3} \pi r^3$ (radius cubed). Mixing these up gives an answer off by a factor of $r$ .

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Cubic unit conversion errors

$1$ ft $= 12$ in, so $1$ ft³ $= 12^3 = 1728$ in³. NOT 12 in³. To convert volume between units, cube the linear factor.

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Misreading a problem about diameter vs radius

Volume formulas use radius. If the problem gives a diameter, divide by 2 first. Sphere with diameter 8 has radius 4, not 8.

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Key takeaways

Box: $V = lwh$ . Cube: $V = s^3$ .
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Cylinder: $V = \pi r^2 h$ . Sphere: $V = \frac{4}{3} \pi r^3$ .
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Cone: $V = \frac{1}{3} \pi r^2 h$ . Pyramid: $V = \frac{1}{3} \cdot \text{base} \cdot h$ .
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Cone/pyramid = one-third of the cylinder/prism with the same base and height.
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Doubling all linear dimensions multiplies volume by $2^3 = 8$ . Cubic conversions cube the factor.
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Where this lesson fits

Before

Area and Perimeter→

Volume

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→Similar Triangles