The volume of a cube is 512

October 20, 2026

2020 Handbook

2020 Test B Math · Question 34 of 57

Question

The volume of a cube is 512 cubic centimeters. What is the length, in centimeters, of one side of this cube?

A8
B $42\frac{2}{3}$
C $85\frac{1}{3}$
D128

Show the answer

Correct answer: A

The volume of a cube is $V=s^{3}$ , where s is the length of one edge.

So $512=s^{3}$ , and $s=\sqrt[3]{512}$ .

If you are not sure how to find the cube root, you can estimate.

We know $10\times 10\times 10=10^{3}=1{,}000$ , so $\sqrt[3]{1{,}000}=10$ .

Since 512 is less than 1,000, the answer must be less than 10. The only given option less than 10 is 8. Double-check to make sure this is correct: $8^{3}=8\times 8\times 8=512$ .

Question analysis

Given only that a cube holds 512 cubic centimeters, students must find the length of one side, running the volume formula backward. That calls for a cube root, since the usual volume problem runs from length to volume. The question says “side” where a later printing says “edge”; both mean the length of one of the cube's edges.

NYS standards: 8.EE.2 square and cube roots
6.G.5 perfect squares and perfect cubes
Grade level: Grade 8 content, by NYS standard
Compared with grade 7: Above grade 7 Grade 6 and 7 volume problems start from edge lengths. This one starts from the volume and needs a cube root, a grade 8 skill, for a number above the cubes students must memorize, though only one short computation is needed.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Something you must already know moderate
Hard words minor
Solution: 2 steps
Reading level: Question and choices: about grade 5, 22 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

This is a one-fact question run backward: a cube's volume is one edge length used three times as a factor, so the edge is the number whose cube is 512. The choices help, since 10 × 10 × 10 = 1,000 rules out everything above 10 and leaves one candidate. The wrong choices divide 512 by 12, 6 or 4, counts tied to a cube's edges, its faces or a square's sides, so they catch students who reach for division instead of the formula.

Something you must already know moderate Working it out
Neither V = s³ nor the cube root of 512 is supplied, so students must recall that a cube's volume is edge × edge × edge and find, by memory or trial, that 8 × 8 × 8 = 512. (Anderson & Krathwohl, 2001)
Hard words minor Understanding the question
This printing asks for “one side” of a cube, and in everyday use a cube's sides are its faces; that is a reading hurdle, not extra math, since an edge is meant. (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: Take “side” to mean an edge of the cube, express 512 as one length multiplied by itself three times, and find that length, confirming by multiplying it out.

Traps

Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

B Wrong formula (King et al., 2004)
Reveals a misconception Expected pull: weak
Divides 512 by 12, the cube's number of edges, as if the volume were the total length of all the edges, getting 42 2/3.
C Wrong formula (King et al., 2004)
Reveals a misconception Expected pull: moderate
Reads “side” as one of the six faces and divides 512 by 6, treating the volume like a surface area shared among the faces, which gives 85 1/3.
D Wrong formula (King et al., 2004)
Reveals a misconception Expected pull: moderate
Divides 512 by 4, the step that finds a square's side from its perimeter, instead of looking for the number whose cube is 512.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a cube: $V = s^3$; V is the volume, the space inside, in cubic units. s is the length of one edge. s³ means s × s × s. A cube’s length, width and height are all the same edge, so length × width × height becomes s × s × s.
Cube root: $\text{if } s^3 = V \text{, then } s = \sqrt[3]{V}$; The cube root of V is the number that gives V when you multiply it by itself three times. It undoes cubing. For example, 2 × 2 × 2 = 8, so the cube root of 8 is 2.

Sort out the question
The cube’s volume is 512 cubic centimeters. Volume is the space inside a solid. You need the length of one side.
Here a side means an edge, one of the 12 straight lines where two faces meet. All 12 edges of a cube are the same length, so one edge tells you all of them.
Write the volume formula
$V = s^3$
$512 = s \times s \times s$
You need a number that makes 512 when you multiply it by itself three times. That number is the cube root of 512.
$\text{if } s^3 = V \text{, then } s = \sqrt[3]{V}$
$s = \sqrt[3]{512}$
Try numbers to find the edge
Start with an easy number.
$10 \times 10 \times 10 = 1{,}000$
That is too big, so the edge is less than 10. Only one choice, 8, is less than 10. Try it.
$8 \times 8 \times 8 = 64 \times 8 = 512$
It works. Each edge is 8 centimeters.
Rule out the other choices
Watch out: in everyday talk, a cube’s sides can mean its 6 flat faces. But the volume is not shared out among the faces. Dividing 512 by 6 gives $85\frac{1}{3}$ , choice C. Dividing by the 12 edges gives $42\frac{2}{3}$ , choice B. Dividing by 4 gives 128, choice D.
So the answer is A, 8 centimeters.

Check your answer

Check the units. An edge is a length, in centimeters. Cubing it gives cubic centimeters: $8^3 = 512$ . And any choice of 10 or more makes a volume of at least 1,000 cubic centimeters, far too big.

The original DOE question

Question 91 as printed on page 196 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2021 Admissions, page 196, question 91. Open page 196 of the handbook (PDF)