A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
Try it yourself first — the explanation is most useful after you’ve attempted it.
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Toolkit + CCSS Solution
Understand
Restated: A solid 3-inch cube has a 2-by-2 square hole punched straight through the center of each face, with the cut edges parallel to the cube's edges. Find the volume of the solid that is left.
Givens: The starting solid is a cube with side length 3 inches.; Each face gets a 2-inch by 2-inch square hole cut in its center.; Every hole is drilled straight through the cube, parallel to the cube's edges.; Opposite faces share the same tunnel, so there are 3 tunnels in all (one per pair of opposite faces).
Unknowns: The volume, in cubic inches, of the remaining solid.
Understand
Restated: A solid 3-inch cube has a 2-by-2 square hole punched straight through the center of each face, with the cut edges parallel to the cube's edges. Find the volume of the solid that is left.
Givens: The starting solid is a cube with side length 3 inches.; Each face gets a 2-inch by 2-inch square hole cut in its center.; Every hole is drilled straight through the cube, parallel to the cube's edges.; Opposite faces share the same tunnel, so there are 3 tunnels in all (one per pair of opposite faces).
Plan
Primary tool: #16 Change Focus / Count the Complement
The leftover solid is an awkward frame shape, but the material that was removed is made of simple boxes. So instead of measuring what stays, measure what is taken away and subtract it from the full cube. The catch is that the three tunnels overlap, so counting the removed boxes is its own subproblem that needs care about the shared center.
Execute — Answer: A
#16 Change Focus / Count the Complement 5.MD.C.5Step 1
Volume of the full cube
Start with the whole cube before any holes.
A cube of side 3 has volume 3 times 3 times 3.
$$3 \times 3 \times 3 = 27$$
💡 Everything left over must come out of these 27 cubic inches, so it is the number to subtract from.
💡 Adding the tunnels overcounts the shared center, so you take the overlaps out and put the once-removed center back exactly once.
#16 Change Focus / Count the Complement 5.MD.C.5Step 5
Subtract to get what remains
The remaining solid is the full cube minus everything that was drilled away: 27 minus 20 equals 7 cubic inches.
So the answer is (A).
$$27 - 20 = 7$$
💡 What is left is simply the starting cube with the removed volume taken out.
[1]
#16 5.MD.C.5Start with the whole cube before any holes. A cube of side 3 has volume 3 times
[2]
#17 5.MD.C.5Each hole is a 2-by-2 square that runs all the way through the length 3 of the c
[3]
#7 7.G.B.6There is one tunnel for each pair of opposite faces, so three tunnels in all. Ea
[4]
#7 5.OA.A.1Use inclusion-exclusion. Add the three tunnels: 3 times 12. Each of the three pa
[5]
#16 5.MD.C.5The remaining solid is the full cube minus everything that was drilled away: 27
Review
Reasonableness: Build the leftover frame directly and check it matches. What survives is 8 corner cubes, each 0.5 by 0.5 by 0.5 (volume 0.125, total 1), plus 12 edge bars, each 2 by 0.5 by 0.5 (volume 0.5, total 6). Adding gives 1 + 6 = 7, the same as 27 minus 20. Also, 7 is the smallest choice, which fits a cube that has been hollowed out almost completely, leaving only a thin frame.
Alternative: Instead of subtracting the removed boxes, add up the pieces that stay. The frame is made of 8 tiny corner cubes and 12 edge bars: 8 times 0.125 plus 12 times 0.5 equals 1 plus 6 equals 7. This avoids overlap bookkeeping entirely because the corner and edge pieces never overlap.
CCSS standards used (min grade 7)
5.MD.C.5 Relate volume to the operations of multiplication and addition (Computing the volume of the whole cube (3 by 3 by 3) and of each rectangular tunnel (2 by 2 by 3).)
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Combining the inclusion-exclusion terms 3 times 12 minus 3 times 8 plus 8 in the correct order.)
7.G.B.6 Solve real-world problems involving area, surface area, and volume (Finding the volume of a composite solid whose three tunnels overlap, correcting for the shared central region rather than just adding.)
⭐ When holes overlap, don't just add up what you drill out; count the shared part once so you don't erase too much.
⭐ When holes overlap, don't just add up what you drill out; count the shared part once so you don't erase too much.
