Question

A plane passes through three vertices of a cube and divides the cube into two parts, a green part and a blue part, and they remain together, as shown below. Eight such cubes are assembled to create a larger cube, where blue portion is on the inside as shown on the right. Calculate the volume of blue part in the larger cube, if the edge of the original cube is 1 cm.

Hint:

A plane cutting through three adjacent vertices of a unit cube always cuts off a tetrahedron of volume $\frac{1}{6}\text{ units}^3$. Combining 8 such tetrahedra at a single point forms a regular octahedron of volume $\frac{4}{3}\text{ units}^3$.

Answer 1

Correct Answer: 3

Step 1: Understanding the Question:
We have a unit cube with an edge length of $1\text{ cm}$.
A plane cuts through three vertices of this cube, separating it into a green part and a blue part.
The blue portion is a corner pyramid (tetrahedron).
Eight such cubes are joined to form a larger $2\text{ cm} \times 2\text{ cm} \times 2\text{ cm}$ cube.
The blue parts are placed facing the center, forming a regular octahedron on the inside.
We need to calculate the total volume of this blue part.

Step 2: Key Formula or Approach:
1. Volume of a corner pyramid (tetrahedron) cut from a cube of edge length $a$:
\[ V_{\text{corner}} = \frac{1}{6} a^3 \]
2. Since eight such cubes are used, the total volume of the blue portion is:
\[ V_{\text{blue}} = 8 \times V_{\text{corner}} \]

Step 3: Detailed Explanation:
1. Calculate the volume of the corner tetrahedron in one cube:
- The plane passes through three mutually adjacent vertices of a corner.
- This forms a right triangular pyramid where the three mutually perpendicular edges are each $1\text{ cm}$ long.
- The base is a right-angled triangle with area:
\[ A_{\text{base}} = \frac{1}{2} \times 1 \times 1 = 0.5\text{ cm}^2 \]
- The height of the pyramid is $1\text{ cm}$.
- The volume of this pyramid is:
\[ V_{\text{corner}} = \frac{1}{3} \times A_{\text{base}} \times \text{height} = \frac{1}{3} \times 0.5 \times 1 = \frac{1}{6}\text{ cm}^3 \]
2. Calculate the combined volume of the 8 blue corners:
- Eight identical cubes are assembled together.
- The total volume of the blue portion (the interior octahedron) is the sum of the volumes of these 8 corner pyramids:
\[ V_{\text{blue}} = 8 \times \frac{1}{6}\text{ cm}^3 = \frac{8}{6} = \frac{4}{3}\text{ cm}^3 \approx 1.33\text{ cm}^3 \]

Step 4: Final Answer:
The volume of the blue part in the larger cube is 1.33 $\text{cm}^3$ (or $\frac{4}{3}$ $\text{cm}^3$).