Asked by unowen
A unit cube with side lengths of 1 unit is cut into cuboids with heights of 1/2, 1/3, and 1/6 units, respectively. The pieces are then placed adjacent to each other to form a staircase.
The total surface area of the original unit cube was 6 units. What is the total surface area of the the staircase made from slices of the original cube?
The total surface area of the original unit cube was 6 units. What is the total surface area of the the staircase made from slices of the original cube?
Answers
Answered by
Steve
I'll assume that the surfaces lying on the ground are not counted. They are all 1x1 anyway, for an additional 3 u^2 area.
The 1/6 block has its top and 3 faces exposed: 1+3*(1/6) = 3/2
The 1/3 block has its top, two sides, and half of another side exposed: 1+2*(1/3)+(1/6) = 11/6
The 1/2 block has its top, two sides, the outside face and 1/3 of its inside face: 1+2*(1/2)+(1/2)+(1/3)(1/2) = 8/3
I was surprised at the answer!
The 1/6 block has its top and 3 faces exposed: 1+3*(1/6) = 3/2
The 1/3 block has its top, two sides, and half of another side exposed: 1+2*(1/3)+(1/6) = 11/6
The 1/2 block has its top, two sides, the outside face and 1/3 of its inside face: 1+2*(1/2)+(1/2)+(1/3)(1/2) = 8/3
I was surprised at the answer!
Answered by
unowen
Figured it out. Total surface area is 9. Top and bottom surfaces DO count, which makes 6 total. The sides are 1/6 and 2 x 1/6, 1/6 + 2 x 1/3, and 1/6 + 2 x 1/2=3 more
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