A wooden box has a square base

2018 Handbook

2018 Test B Math · Question 39 of 57

Also in 2026 Practice Test B Math, question 33; 2025 Practice Test B Math, question 39.

Question

A wooden box has a square base. The height of this box is 3 times the length of one side of the base. If one side of the base is 3 feet long, what is the volume of this box?

A9 cu ft
B27 cu ft
C36 cu ft
D81 cu ft

Show the answer

Correct answer: D

One side of the square base is 3 ft long. Since the height of the box is 3 times the length, then the height is

3 × 3 = 9 ft.

The volume of a rectangular prism is length × height × width. The volume of the wooden box is

V= 3 × 3 × 9 = 81 cu ft.

Question analysis

A wooden box has a square base, and its height is 3 times the length of a side of the base; students find its volume in cubic feet. It tests the volume of a rectangular prism with dimensions described in words. The work is short, but the height must be built from the side length before multiplying.

NYS standards: 6.G.2 volume with fractional edge lengths
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Volume of a box is grade 5 and 6 content, and this is a two-step version with whole numbers. Grade 7 volume problems usually involve composite or triangular prisms, so this sits below.
Estimated difficulty: Routine: one familiar step, little to misread.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Turning words into math minor
Something you must already know minor
Solution: 2 steps
Reading level: Question and choices: about grade 1, 46 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 routine volume question once all three dimensions are written down. Errors come from treating the box as a 3-foot cube, from stopping at the base area, or from mixing perimeter into the product. Listing the dimensions 3, 3 and 9 before multiplying prevents each of them.

Turning words into math minor Understanding the question
"The height of this box is 3 times the length of one side of the base" must become a height of 9 feet before the volume formula can be used. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Something you must already know minor Working it out
The volume of a box, length times width times height, is not given, and the square base means the length and width are both 3 feet. (Anderson & Krathwohl, 2001)

Strategy: Write down all three dimensions first, computing the height from the base side, then multiply length, width and height.

Traps

Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: moderate
Stops at the area of the square base, 3 × 3 = 9, or at the 9-foot height, without bringing in the third dimension.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: strong
Uses 3 feet for the height too, as if the box were a cube, 3 × 3 × 3 = 27; multiplying only 3 × 9 also gives 27.
C Wrong formula (King et al., 2004)
Reveals a misconception Expected pull: weak
Multiplies the base perimeter, 4 × 3 = 12 feet, by 3, putting a perimeter where the base area belongs.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a box: $V = l \times w \times h$; V is the volume, the space inside the box. l is the length, w is the width and h is the height. l times w is the area of the bottom. Multiplying by h stacks that many layers of cubes.

Sort out what you know
The box has a square base with sides of 3 feet. A square has equal sides, so the length and width are both 3 feet. The height is “3 times the length of one side of the base”. You need the volume, the space inside the box.
Find the height
The height is 3 times the 3-foot side.
$h = 3 \times 3 = 9$
The height is 9 feet.
Watch out: the height is not 3 feet. If you use 3 for the height, you get $3 \times 3 \times 3 = 27$ . That is choice B.
Find the volume
Use the volume formula with all three sizes.
$V = 3 \times 3 \times 9 = 81$
The volume is 81 cubic feet.
Watch out: $3 \times 3 = 9$ is only the area of the bottom. Stopping there gives choice A.
Give the answer
The box is 3 feet long, 3 feet wide and 9 feet tall. So the answer is D, 81 cu ft.

Check your answer

Think in layers. The bottom layer holds $3 \times 3 = 9$ cubes, each 1 foot on a side. The box is 9 feet tall, so it holds 9 layers. Then $9 \times 9 = 81$ cubic feet. The units are cubic feet because the answer is a volume.

The original DOE question

Question 96 as printed on page 160 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 160, question 96. Open page 160 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).