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?
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!
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
☺☺☺☺
To find the surface area of the staircase made from slices of the original cube, we need to calculate the surface area of each cuboid and add them up.
Let's start by finding the surface area of each cuboid:
Cuboid 1 (height 1/2): The two larger faces of this cuboid will have the same surface area as the original unit cube. Since the height of this cuboid is 1/2 unit, the area of each of the two larger faces will be (1 unit) * (1/2 unit) = 1/2 square units. The four smaller faces will be the sides of the cuboid, each with dimensions 1 unit by 1/2 unit, so the area of each of these faces will be (1 unit) * (1/2 unit) = 1/2 square units. Therefore, the surface area of this cuboid is (1/2 square units * 2) + (1/2 square units * 4) = 3 square units.
Cuboid 2 (height 1/3): Again, the two larger faces of this cuboid will have the same surface area as the original unit cube. Since the height of this cuboid is 1/3 unit, the area of each of the two larger faces will be (1 unit) * (1/3 unit) = 1/3 square units. The four smaller faces will be the sides of the cuboid, each with dimensions 1 unit by 1/3 unit, so the area of each of these faces will be (1 unit) * (1/3 unit) = 1/3 square units. Therefore, the surface area of this cuboid is (1/3 square units * 2) + (1/3 square units * 4) = 2 square units.
Cuboid 3 (height 1/6): Similarly, the two larger faces of this cuboid will have the same surface area as the original unit cube. Since the height of this cuboid is 1/6 unit, the area of each of the two larger faces will be (1 unit) * (1/6 unit) = 1/6 square units. The four smaller faces will be the sides of the cuboid, each with dimensions 1 unit by 1/6 unit, so the area of each of these faces will be (1 unit) * (1/6 unit) = 1/6 square units. Therefore, the surface area of this cuboid is (1/6 square units * 2) + (1/6 square units * 4) = 1 square unit.
Now, we can find the total surface area of the staircase by adding up the surface area of each cuboid: 3 square units + 2 square units + 1 square unit = 6 square units.
Therefore, the total surface area of the staircase made from slices of the original unit cube is 6 square units, which is the same as the surface area of the original unit cube.