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Practice Test

6.G.2 volume with fractional edge lengths

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Question 1 of 7

2026 Practice Test C Math, question 42

Question

Tariq is making a clay block in the shape of a cube with a side length of 20 cm20\text{ cm}. He hollows out the center by removing clay to form a rectangular prism-shaped hole with a base of 5 cm⋅5 cm5\text{ cm} \cdot 5\text{ cm} and a height of 20 cm20\text{ cm}.

A cube labeled 20 cm along its front right edge. A tall rectangular prism-shaped hole runs from the top face of the cube down to the bottom face, the full height of the cube; its square opening on the top face is shaded gray and labeled 5 cm across. Hidden edges of the cube and the hole are dashed.

What is the volume, in cubic centimeters, of the block after the clay is removed from the center?

Enter your answer in the space.

Fill-in question: work out the answer and type it as a number.

Show the answer

Correct answer: 7,500

First, find the volume of the cube using the formula V=lwhV = lwh.

The length, width, and height of the cube are each equal to 20 cm.

Vc=20 cm×20 cm×20 cm=8,000 cm3V_c = 20\text{ cm} \times 20\text{ cm} \times 20\text{ cm} = 8{,}000\text{ cm}^3

Next, find the volume of the cut-out square prism, with equal length and width of 5 cm and height of 20 cm.

Vp=5 cm×5 cm×20 cm=500 cm3V_p = 5\text{ cm} \times 5\text{ cm} \times 20\text{ cm} = 500\text{ cm}^3

Finally, subtract the volume of the prism from the volume of the cube.

8,000 cm3−500 cm3=7,500 cm38{,}000\text{ cm}^3 - 500\text{ cm}^3 = 7{,}500\text{ cm}^3

Question analysis

Tariq makes a clay cube 20 centimeters on each side and removes a rectangular prism from its center, with a 5 cm by 5 cm base and a height of 20 cm. Students type the volume of the block that is left, in cubic centimeters. It tests the volume of rectangular prisms, and the hole's size must be read carefully from the words and figure.

NYS standards
  • 7.G.6 area, surface area, and volume
  • 6.G.2 volume with fractional edge lengths
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Finding the volume of a solid made from rectangular prisms, here by taking one away, is named in the grade 7 standards. This is a direct case with whole numbers, two volumes and one subtraction.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 8, 66 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 direct composite-volume problem with whole numbers: one cube, one hole, one subtraction. What it checks is that both volumes are real volumes, the hole as tall as the cube, and that the subtraction is done at the end. Students who stop at the hole’s volume, or who subtract only the square face, give amounts that are easy to type and hard to doubt without choices.

  1. Something you must already know minor Working it out

    The volume of a rectangular prism, length times width times height, is not given, and it is needed twice, once for the 20 cm cube and once for the hole. (Anderson & Krathwohl, 2001)

  2. Reading a figure, table or graph minor Understanding the question

    The figure labels only the 5 cm opening and the cube’s 20 cm edge; the hole’s 20 cm height comes from the words, while the drawing shows it reaching from top to bottom. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  3. An answer in an unexpected form minor Choosing the answer

    The volume is typed with no choices to compare, so the hole’s 500 cubic centimeters or the cube’s 8,000 can be entered without any warning. (Enright, Morley & Sheehan, 2002)

Strategy: Find the volume of the full cube, then the volume of the square hole that runs from the top face to the bottom, and subtract the hole from the cube.

Traps

  • Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
  • Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
  • Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a rectangular prism
V=l×w×hV = l \times w \times h
V is the volume, the space inside a solid. l is the length, w is the width and h is the height. A cube is a prism with all three the same. Volume is measured in cubic centimeters, tiny cubes 1 cm on each side.

  1. Sort out the question

    Tariq’s block starts as a cube 20 cm on each side. He removes a hole shaped like a rectangular prism, a box shape with flat rectangle sides. Its base is 5 cm by 5 cm, and its height is 20 cm. The figure shows it running from the top face to the bottom face.

    You need the volume of the clay that is left, in cubic centimeters.

  2. Find the volume of the cube

    Use the volume formula. The length, width and height are all 20 cm:

    V=20×20×20=8,000V = 20 \times 20 \times 20 = 8{,}000

    The full cube holds 8,000 cubic cm of clay.

  3. Find the volume of the hole

    The hole is a prism too. Its length and width are 5 cm, and its height is 20 cm:

    V=5×5×20=500V = 5 \times 5 \times 20 = 500

    So 500 cubic cm of clay are removed.

    Watch out: the hole is 20 cm tall, not 5 cm. Treating it as a small 5 cm cube gives 125 and the wrong answer 7,875. And 5 × 5 = 25 is only the area of the opening, not a volume. Subtracting it gives 7,975, also wrong.

  4. Subtract the hole from the cube

    The clay left is the cube minus the hole:

    8,000−500=7,5008{,}000 - 500 = 7{,}500

    The block has 7,500 cubic cm of clay left.

    Watch out: 500 is the clay taken out, not the clay that is left.

  5. Enter the number

    Type only the number, with no units. You can type 7500 or 7,500.

    So you would enter 7,500.

Check your answer

Think of the cube as 20 flat layers, each 1 cm thick. Each layer is a 20 by 20 square, 400 square cm, with a 5 by 5 hole, 25 square cm. So each layer holds 400−25=375400 - 25 = 375 cubic cm of clay. Then 20×375=7,50020 \times 375 = 7{,}500. It matches.

More questions like this

The original DOE question
Question 42 as shown on the DOE’s digital test, page 115 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test C Math, page 115, question 42.

Question 2 of 7

2024 Test B Math, question 51

Question

Rita used 125 small solid cubes to build a large solid cube. The measure of each edge of the small cubes equals 1 inch. What is the measure of each edge of the large cube, in inches?

  • A5
  • B25
  • C41.33
  • D125
Show the answer

Correct answer: A

The volume of a cube with a side length of s is

V=s3V=s^{3}

Since 125 small cubes were used to build the large cube, 125=s3125=s^{3}.

Each side of the large cube is made up of 5 small cubes.

125=53125=5^{3}

Since 5 small cubes make up each edge of the large cube, and the edge of each small cube measures 1 inch, the measure of each edge of the large cube is 1×5=5 inches1\times 5=5\text{ inches}.

Question analysis

Rita stacks 125 one-inch cubes into a single large solid cube. Students find the length of each edge of the large cube in inches. It tests volume and perfect cubes with a building-block model, and the care needed is in telling apart the cubes along an edge, across a face and in the whole solid.

NYS standards
  • 6.G.5 perfect squares and perfect cubes
  • 6.G.2 volume with fractional edge lengths
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Building a large cube from unit cubes is the model grade 6 uses to introduce perfect cubes. It is a single step with small numbers, easier than typical grade 7 geometry, which works with composite solids and formulas.
Estimated difficulty
Routine: one familiar step, little to misread.
What makes it hard
Solution
1 step
Reading level
Question and choices: about grade 3, 35 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 one of the easiest questions on the test, since 125 is a familiar cube and the small cubes make the large one easy to picture as 5 rows of 5 in 5 layers. The wrong choices name a layer instead of an edge, a division by 3 for three dimensions, or the total count itself, and multiplying 5 × 5 × 5 as a check rules all of them out.

  1. Something you must already know minor Working it out

    Students must see that 125 one-inch cubes make a volume of 125 cubic inches, and know or find that 5 × 5 × 5 = 125. (Anderson & Krathwohl, 2001)

Strategy: Think of the large cube as equal rows, columns and layers of small cubes, and find the number that, used three times as a factor, gives 125.

Traps

  • Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a cube
V=s3V = 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 s, so length × width × height becomes s × s × s.

  1. Sort out the question

    Rita stacks 125 small cubes into one big solid cube. Each small cube has edges of 1 inch. You need the length of one edge of the big cube, in inches.

  2. Find the big cube’s volume

    Each small cube is 1×1×1=11 \times 1 \times 1 = 1 cubic inch. The big cube is solid, with no gaps. So its volume is 125 cubic inches.

  3. Find the edge length

    V=s3V = s^3

    125=s×s×s125 = s \times s \times s

    Find the number that makes 125 when you use it three times. Try 5.

    5×5×5=25×5=1255 \times 5 \times 5 = 25 \times 5 = 125

    So 5 small cubes sit along each edge. Each one is 1 inch long, so the edge is 5 inches.

  4. Rule out the other choices

    Watch out: 25, choice B, is the number of small cubes in one layer, 5 rows of 5. 125, choice D, is the number of cubes, or the volume. Dividing 125 by 3 does not give an edge either. That is about 41.67, near choice C.

    So the answer is A, 5.

Check your answer

Picture building it. A row of 5 cubes is 5 inches long. Five rows make a layer of 25 cubes. Five layers make 25×5=12525 \times 5 = 125 cubes. That matches.

More questions like this

The original DOE question
Question 108 as printed on page 213 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 213, question 108. Open page 213 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).

Question 3 of 7

2023 Test A Math, question 50

Question

A rectangular concrete driveway is 30 feet long, 8 feet wide, and 6 inches thick. What is the volume of the concrete?

  • A44 cu ft
  • B48 cu ft
  • C120 cu ft
  • D240 cu ft
Show the answer

Correct answer: C

First convert 6 inches to 0.5 foot. Then, calculate the volume, in cubic feet:

30×8×0.5=12030 \times 8 \times 0.5 = 120

Question analysis

This volume word problem describes a concrete driveway 30 feet long, 8 feet wide and 6 inches thick, and asks for the volume of the concrete in cubic feet. Students convert inches to feet and multiply three dimensions. Mixing units, or stopping at the area of the driveway's surface, is the main way to go wrong.

NYS standards
  • 6.G.2 volume with fractional edge lengths
  • 6.RP.3d converting measurement units
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Volume of a rectangular prism with a fractional edge is grade 6 content. The only extra step is changing inches to feet before multiplying, so it is lighter than grade 7 volume work with composite or triangular prisms.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
2 steps
Reading level
Question and choices: about grade 2, 27 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

Volume of a box is familiar, so the question is decided by the mix of units: two dimensions in feet and one in inches. Multiplying the three numbers as printed gives 1,440, which is not a choice, and that should send students back to convert. The listed wrong answers instead come from leaving out the thickness or combining the numbers without the volume rule.

  1. Something you must already know minor Working it out

    The stem gives no formula and no conversion, so students must bring both the volume rule, length times width times height, and the 12 inches in a foot. (Anderson & Krathwohl, 2001)

  2. Error-prone arithmetic minor Working it out

    The thickness becomes the fraction 1/2 foot, so the product is 30 × 8 × 1/2, or, if the inches are kept, 1,440 divided by 12. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Change 6 inches to a fraction of a foot first so that all three measurements share a unit, then multiply length, width and thickness.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Inches to feet
feet=inches÷12\text{feet} = \text{inches} \div 12
There are 12 inches in 1 foot. So to change inches into feet, divide the number of inches by 12.
Volume of a rectangular box
V=l×w×hV = l \times w \times h
V is the volume, the space inside a box shape. l is the length, w is the width and h is the height, here the thickness. All three must use the same unit. The answer is in cubic units, such as cubic feet.

  1. Sort out the question

    The driveway is a flat box of concrete. It is 30 feet long, 8 feet wide and 6 inches thick. You need its volume, the space the concrete fills, in cubic feet. The thickness is in inches, so change it to feet first.

  2. Change the thickness to feet

    feet=inches÷12\text{feet} = \text{inches} \div 12

    6÷12=126 \div 12 = \frac{1}{2}

    So 6 inches is half a foot. Now all three measures are in feet.

  3. Multiply the three measures

    V=l×w×hV = l \times w \times h

    V=30×8×12=240×12=120V = 30 \times 8 \times \frac{1}{2} = 240 \times \frac{1}{2} = 120

    The volume is 120 cubic feet. Watch out: 30 × 8 = 240 is only the area of the top surface, in square feet. Stopping there gives choice D. Adding 30 + 8 + 6 = 44 gives choice A.

    So the answer is C, 120 cu ft.

Check your answer

Think of it another way. A slab 30 feet by 8 feet and a full foot thick would hold 240 cubic feet. This one is only half a foot thick, so it holds half as much, 120 cubic feet. Using 6 instead of one half would give 1,440, far too big.

More questions like this

The original DOE question
Question 107 as printed on page 89 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 89, question 107. Open page 89 of the handbook (PDF)

Question 4 of 7

2023 Test B Math, question 24

Question

A prism is made up of 9 congruent cubes. Each of the cubes has side lengths measuring 4 units. What is the volume, in cubic units, of the prism?

  • A108
  • B144
  • C480
  • D576
Show the answer

Correct answer: D

First, determine the volume of each congruent cube. All the side lengths of the cubes are equal, so the length, width, and height of each cube is 4 units. To determine the volume of each cube, multiply length×width×height\text{length} \times \text{width} \times \text{height}:

4×4×4=644 \times 4 \times 4 = 64

So, the volume of each cube is 64 cubic units.

Left: a prism built from 9 cubes, three rows of three, one cube deep, with one cube’s height, width and depth each labeled 4. Right: a single cube whose height, width and depth are each labeled 4.

The prism is comprised of 9 of these cubes, so multiply the volume of each cube (64) by the total number of cubes (9) to determine the volume, in cubic units, of the prism:

64×9=57664 \times 9 = 576

Question analysis

A prism is built from 9 identical cubes, and each cube has edges 4 units long. Students find the volume of the prism in cubic units. It tests the volume of a cube and how volumes add when solids are put together, and it rewards keeping area, surface area and volume apart.

NYS standards
  • 6.G.2 volume with fractional edge lengths
  • 6.G.5 perfect squares and perfect cubes
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Volume of cubes and rectangular prisms is grade 5 and 6 content. This item needs one cube's volume and a multiplication by 9, two short steps, so it is below typical grade 7 work.
Estimated difficulty
Routine: one familiar step, little to misread.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 3, 27 words.
Corrected from the DOE handbook
  • A letter x printed in place of a multiplication sign in the explanation was corrected.
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

The question needs one formula and one multiplication, and the stem does not even say how the cubes are arranged, because the volume does not depend on it. The wrong choices come from mishandling the cube’s volume: the exponent multiplied instead of applied, the area of one face, or a surface area. Writing the volume out as edge times edge times edge, in cubic units, rules all three out.

  1. Something you must already know minor Working it out

    A cube’s volume, edge × edge × edge, is not given; with it, each 4-unit cube holds 64 cubic units, and the arrangement of the 9 cubes does not matter. (Anderson & Krathwohl, 2001)

Strategy: Find the volume of one cube by multiplying its three edges, then multiply by the number of cubes; how the cubes are arranged does not matter.

Traps

  • Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The handbook gives the volume of a cube as “length x width x height”, with the letter x where it means “times”. In a math explanation the letter x reads as a variable, so the site writes length × width × height, the sign the handbook itself uses on the next line (4 × 4 × 4 = 64). Nothing else changes: each cube is 64 cubic units, and the 9 cubes make 64 × 9 = 576.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a cube
V=s×s×sV = s \times s \times s
V is the volume, the space inside a solid, in cubic units. s is the length of one edge. All the edges of a cube are equal, so its length, width and height are all s. Multiplying them counts how many unit cubes fit inside.

  1. Sort out the question

    A prism, a solid with flat faces, is made of 9 congruent cubes. Congruent means exactly the same size and shape. Each cube has sides of 4 units. You need the volume of the whole prism, in cubic units.

    The question doesn’t say how the cubes are stacked. You don’t need to know. Volume is the space inside, and 9 cubes fill the same space however they are arranged.

  2. Find the volume of one cube

    V=s×s×sV = s \times s \times s

    V=4×4×4=64V = 4 \times 4 \times 4 = 64

    Each cube has a volume of 64 cubic units. Watch out: 434^3 means 4×4×44 \times 4 \times 4, not 4×3=124 \times 3 = 12. That slip leads to 108, choice A. Also, 4×4=164 \times 4 = 16 is only the area of one face. It leads to 144, choice B.

  3. Multiply by the number of cubes

    The prism is 9 of these cubes, so its volume is 9 times as big.

    64×9=57664 \times 9 = 576

    The prism’s volume is 576 cubic units.

  4. Give the answer

    Watch out: volume counts the space inside, not the outside faces. Suppose the cubes formed one flat 3-by-3 layer. Adding up the areas of its outside faces gives 480, choice C. That is surface area, not volume. So the answer is D, 576.

Check your answer

Estimate: 64×10=64064 \times 10 = 640, and one cube fewer is 640−64=576640 - 64 = 576. You can also test one arrangement. A flat 3-by-3 layer is 12 by 12 by 4 units, and 12×12×4=57612 \times 12 \times 4 = 576. Same answer.

More questions like this

The original DOE question
Question 81 as printed on page 198 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 198, question 81. Open page 198 of the handbook (PDF) This is the DOE’s version, before the corrections listed in the analysis.

Question 5 of 7

2023 Test B Math, question 49

Question

A rectangular swimming pool of uniform depth is 20 yards long and 14 yards wide. If the water is 6 feet deep, what is the volume of water in the pool, in cubic yards?

  • A40
  • B286
  • C560
  • D1,680
Show the answer

Correct answer: C

First, change 6 feet to yards:

6 feet×1 yard3 feet=2 yards6\text{ feet}\times\frac{1\text{ yard}}{3\text{ feet}}=2\text{ yards}

Then, multiply the dimensions of the pool to find the volume, in cubic yards:

20×14×2=56020\times 14\times 2=560

Question analysis

A rectangular swimming pool is 20 yards long and 14 yards wide, and the water is 6 feet deep. Students find the volume of water in cubic yards. It tests the volume of a rectangular prism, and the depth is in a different unit from the other measurements, so it has to be converted first.

NYS standards
  • 6.G.2 volume with fractional edge lengths
  • 6.RP.3d converting measurement units
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 The volume of a rectangular prism is grade 5 and 6 content, and the conversion here gives a whole number. Grade 7 volume work is usually longer, with composite or triangular prisms; here the only extra step is one unit conversion.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 6, 31 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

The only real trap is the mixed units: 20 and 14 are yards, 6 is feet, and the answer is wanted in cubic yards. Multiplying the three numbers as printed gives a listed choice three times too large. Underlining the unit of each measurement before multiplying, and converting the odd one out, takes seconds.

  1. Combines ideas minor Working it out

    The length and width are in yards but the depth is in feet, so a conversion, 6 feet = 2 yards, must come before the volume formula is used. (Embretson, 1998; Daniel & Embretson, 2010)

  2. Something you must already know minor Working it out

    Neither 3 feet = 1 yard nor volume = length × width × depth is given; both are assumed in a stem that asks for cubic yards. (Anderson & Krathwohl, 2001)

Strategy: Change the depth from feet to yards first, so that all three measurements are in yards, then multiply the length, width and depth.

Traps

  • Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
  • Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
  • Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a rectangular box
V=l×w×hV = l \times w \times h
V is the volume, the space inside, in cubic units. l is the length, w is the width, and h is the height or depth. All three must be in the same unit. Multiplying them counts the unit cubes that fill the box.
Feet to yards
yards=feet÷3\text{yards} = \text{feet} \div 3
There are 3 feet in 1 yard. So to change feet to yards, divide the number of feet by 3. A yard is longer than a foot, so you get a smaller number.

  1. Sort out the measurements

    The pool is shaped like a box. Uniform depth means the water is equally deep everywhere. The pool is 20 yards long and 14 yards wide. The water is 6 feet deep. You need the volume, the space the water fills, in cubic yards.

  2. Change the depth to yards

    Two sides are in yards, but the depth is in feet. All three must use the same unit before you multiply. The answer is in cubic yards, so change feet to yards.

    6÷3=26 \div 3 = 2

    The water is 2 yards deep.

  3. Find the volume

    Use the volume formula with all three measurements in yards.

    V=20×14×2=560V = 20 \times 14 \times 2 = 560

    The pool holds 560 cubic yards of water.

  4. Give the answer

    Watch out: Multiplying 20 × 14 × 6 mixes yards with feet. It gives 1,680, choice D, which is three times too big.

    So the answer is C, 560 cubic yards.

Check your answer

Check the units. Yards times yards times yards gives cubic yards, just what the question asks for. Also, 1,680 ÷ 3 = 560. So using 6 feet unchanged makes the answer exactly three times too big.

More questions like this

The original DOE question
Question 106 as printed on page 205 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 205, question 106. Open page 205 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).

Question 6 of 7

2022 Test B Math, question 1

Question

A cube whose right edge is labeled 18 mm. A hole shaped like a square prism runs straight down through the middle of the cube from the top face to the bottom face; its square opening on the top face is shaded gray, and one side of the opening is labeled 6 mm. The hidden edges of the cube and of the hole are dashed.

Beth makes a bead in the shape of a cube with side length 18 millimeters. She cuts a hole in the shape of a square prism out of the center of the cube so that the bead will fit on a string. The base of the square prism has a side length of 6 millimeters. After the hole is cut out, what is the volume of the bead in cubic millimeters?

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: 5184

First, find the volume of the cube using the formula V=lwhV = lwh.

The length, width, and height of the cube are equal (18 mm).

Vc=18×18×18=5,832 cubic mmV_{c} = 18 \times 18 \times 18 = 5{,}832\text{ cubic mm}

Next, find the volume of the cut-out square prism, with equal length and width (6 mm), and height of 18 mm.

Vp=6×6×18=648 cubic mmV_{p} = 6 \times 6 \times 18 = 648\text{ cubic mm}

Finally, subtract the volume of the prism from the volume of the cube.

5,832−648=5,184 cubic mm5{,}832 - 648 = 5{,}184\text{ cubic mm}

Question analysis

Beth makes a bead from a cube 18 millimeters on each side and cuts a square prism-shaped hole, 6 millimeters across, through its center for a string. Students find the volume of the bead that is left, in cubic millimeters. It tests the volume of rectangular prisms, and the hole's height is not given in words, so it has to be read from the figure.

NYS standards
  • 7.G.6 area, surface area, and volume
  • 6.G.2 volume with fractional edge lengths
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Finding the volume of a solid made by removing one rectangular prism from another is grade 7 work. This case is direct, but the hole's height is not stated in words, and cubing 18 by hand is heavier arithmetic, typed with no choices.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
3 steps
Reading level
Question and choices: about grade 6, 69 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

The method is short, cube minus hole, but the hole’s height is never stated and has to be read from the drawing. Students who picture the hole as a 6 mm cube take away only 216 cubic millimeters, and those who stop at either volume grid a number that looks finished. Seeing the hole as a square shaft, 6 by 6 by 18, and checking that the bead holds less than the cube catches all three slips.

  1. Reading a figure, table or graph moderate Understanding the question

    The stem gives only the 6 mm side of the hole’s square base; that the hole runs the full 18 mm, from the top face to the bottom, shows only in the figure’s dashed edges. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  2. Something you must already know minor Working it out

    Neither volume formula is given: the cube needs edge × edge × edge, and the square hole needs its 36 square millimeter base times its 18 mm length. (Anderson & Krathwohl, 2001)

  3. Error-prone arithmetic minor Working it out

    Cubing 18 by hand, 324 × 18 = 5,832, and then 6 × 6 × 18 = 648 are multi-digit products, and a carrying slip in either changes the final difference. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  4. An answer in an unexpected form minor Choosing the answer

    The volume is gridded with no choices to compare, so entering the hole’s 648 or the whole cube’s 5,832 brings no warning that a step is missing. (Enright, Morley & Sheehan, 2002)

Strategy: Find the volume of the whole cube, then the volume of the square hole, which runs from the top face to the bottom face, and subtract the hole from the cube.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a rectangular prism
V=l×w×hV = l \times w \times h
V is the volume, the space inside a solid, in cubic units. l is the length, w is the width and h is the height. A cube is a prism whose length, width and height are all equal. Multiplying counts the unit cubes that fill the box, layer by layer.

  1. Sort out the question

    The bead starts as a cube, a box whose edges are all 18 millimeters long. A hole is cut through its middle. The hole is a square prism, a box with a square base. Each side of that square is 6 millimeters.

    You need the volume of the bead that is left, in cubic millimeters. Volume is the space a solid takes up.

  2. Read the hole’s height from the figure

    The words don’t say how tall the hole is. Look at the figure. The gray square sits on the top face. Its dashed edges run all the way down to the bottom face.

    So the hole is as tall as the cube, 18 millimeters. The hole measures 6 by 6 by 18 millimeters.

  3. Find the volume of the cube

    Use the volume formula. All three measures of the cube are 18.

    V=l×w×h=18×18×18=5,832V = l \times w \times h = 18 \times 18 \times 18 = 5{,}832

    Work it in two parts: 18×18=32418 \times 18 = 324, then 324×18=5,832324 \times 18 = 5{,}832. The whole cube is 5,832 cubic millimeters.

  4. Find the volume of the hole

    Use the same formula for the hole: 6 by 6 by 18.

    V=l×w×h=6×6×18=648V = l \times w \times h = 6 \times 6 \times 18 = 648

    The hole is 648 cubic millimeters.

    Watch out: the hole is not a small 6 mm cube. Using 6×6×6=2166 \times 6 \times 6 = 216 would give 5,616, which is wrong.

  5. Subtract the hole from the cube

    The bead is the cube with the hole taken out. So subtract.

    5,832−648=5,1845{,}832 - 648 = 5{,}184

    The bead’s volume is 5,184 cubic millimeters.

    Watch out: 648 is the part cut out, and 5,832 is the cube before the cut. Neither one is the bead. So you would grid 5184.

Check your answer

Picture the cube cut into tall columns, each the size of the hole. It is 3 columns wide and 3 deep, so it holds 3×3=93 \times 3 = 9 columns. The hole takes 1, leaving 8. And 8×648=5,1848 \times 648 = 5{,}184. It matches.

More questions like this

The original DOE question
Question 58 as printed on page 197 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 197, question 58. Open page 197 of the handbook (PDF)

Question 7 of 7

2018 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.
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.

  1. 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)

  2. 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

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Volume of a box
V=l×w×hV = 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.

  1. 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.

  2. Find the height

    The height is 3 times the 3-foot side.

    h=3×3=9h = 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 3×3×3=273 \times 3 \times 3 = 27. That is choice B.

  3. Find the volume

    Use the volume formula with all three sizes.

    V=3×3×9=81V = 3 \times 3 \times 9 = 81

    The volume is 81 cubic feet.

    Watch out: 3×3=93 \times 3 = 9 is only the area of the bottom. Stopping there gives choice A.

  4. 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 3×3=93 \times 3 = 9 cubes, each 1 foot on a side. The box is 9 feet tall, so it holds 9 layers. Then 9×9=819 \times 9 = 81 cubic feet. The units are cubic feet because the answer is a volume.

More questions like this

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).