SAT Math: Area and volume

Area and volume (the reference sheet gives the formulas).

Geometry and Trigonometry · Area and volume
Both modules
14 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise area and volume →Adaptive SAT module

What the test covers

Composite areas; volume scaling by k^3
Reference-sheet formulas

Key ideas

Cylinder $π r^{2} h$ , cone $\frac{1}{3} π r^{2} h$ , sphere $\frac{4}{3} π r^{3}$ , prism = base area $\times$ height.
Scaling every length by $k$ multiplies area by $k^{2}$ and volume by $k^{3}$ .
Composite shapes: add or subtract simple pieces.

Common mistakes

Use the radius, not the diameter.
Check whether the answer is wanted in terms of $π$ .

Do it in Desmos

Use Desmos only for the arithmetic; keep $π$ symbolic if the options do.

IB link: IB SL 3.1.

Worked example

A right circular cone has base radius 5 and height 9. What is its volume?

$225 π$
$135 π$
$\frac{225 π}{2}$
$75 π$

Answer: D: $75 π$

$V = \frac{1}{3} π r^{2} h$ (on the reference sheet) gives $75 π$ .

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

A right circular cylinder has radius 3 and height 12. What is its volume?

$72 π$
$36 π$
$108 π$
$432 π$

Show the answer and solution

Answer: C: $108 π$

$V = π r^{2} h$ (on the reference sheet) gives $108 π$ .

Question 2

Each of the length, width and height of a rectangular box is multiplied by 3. By what factor is the volume multiplied?

$9$
$27$
$3$
$6$

Show the answer and solution

Answer: B: $27$

Volume is a product of three lengths, so it is multiplied by $3 \times 3 \times 3 = 27$ .

Question 3

A solid has volume 8 cubic centimeters. Every length of the solid is multiplied by 3. What is the volume of the new solid, in cubic centimeters?

Student-produced response: type your answer.

Show the answer and solution

Answer: 216

Volume scales by the cube of the length factor: $8 \times 3^{3} = 216$ .