Problem 24 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
area
A cube is cut in three directions as shown, to produce eight cuboids (each cut is parallel to one of the faces of the cube). What is the ratio of the total surface area of the eight cuboids to the surface area of the original cube?
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Answer: D — 2 : 1
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Hint 1 of 2
Each straight cut makes two new faces, each equal to the cross-section it slices.
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Hint 2 of 2
Add the new face area from the three cuts to the original surface and compare.
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Approach: count the new surface from each cut
- Let the cube's surface be 6 (in face units). Each middle cut adds two faces of area 1, i.e. +2 per cut.
- Three cuts add 6, giving total 6 + 6 = 12.
- The ratio of new total to original is 12 : 6 = 2 : 1.
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