+5 Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is 3, then find the volume of the resulting polyhedron.
+136 To find the volume of the resulting polyhedron after Donatello slices off a pyramid from each corner of the marble cube, we can approach the problem step-by-step.
### Solution By Steps
**Step 1: Determine the Volume of the Original Cube**
The side length of the original cube is given as \( s = 3 \). The volume \( V_{\text{cube}} \) of a cube with side length \( s \) is:
\[
V_{\text{cube}} = s^3 = 3^3 = 27
\]
**Step 2: Understand the Structure of the Sliced Cube (a Truncated Cube)**
When Donatello slices off a pyramid from each corner of the cube such that all edges of the resulting polyhedron are equal, he effectively transforms the cube into a truncated cube. This truncated cube has:
1. \(8\) triangular faces (where each corner was sliced off).
2. \(6\) octagonal faces (original cube faces, now truncated).
In a truncated cube of this type, each corner slice removes a smaller triangular pyramid with a base that is an equilateral triangle and with edges that now match the truncated polyhedron's edge length.
**Step 3: Calculate the Edge Length of the Truncated Cube**
Since the edge length of the truncated polyhedron is made equal after truncation, each new edge length is equal to the distance from the
original cube corner to the point where the slice begins. This distance is \( \frac{s}{3} = \frac{3}{3} = 1 \).
Thus, the edge length of the truncated cube is 1.
**Step 4: Calculate the Volume of Each Corner Pyramid Removed**
Each of the 8 corners has a pyramid sliced off with an edge length of 1. Since these pyramids are symmetric and identical:
1. The height of each triangular pyramid is also 1 (the distance from the original corner to the truncated plane).
2. The base area of each pyramid is the area of an equilateral triangle with side length 1:
\[
\text{Base Area} = \frac{\sqrt{3}}{4} \times 1^2 = \frac{\sqrt{3}}{4}
\]
The volume \( V_{\text{pyramid}} \) of each triangular pyramid is:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times \frac{\sqrt{3}}{4} \times 1 = \frac{\sqrt{3}}{12}
\]
**Step 5: Calculate the Total Volume Removed**
Since there are 8 pyramids removed, the total volume removed is:
\[
V_{\text{removed}} = 8 \times \frac{\sqrt{3}}{12} = \frac{2\sqrt{3}}{3}
\]
**Step 6: Calculate the Volume of the Resulting Polyhedron**
The volume of the resulting polyhedron \( V_{\text{truncated cube}} \) is the volume of the original cube minus the total volume removed:
\[
V_{\text{truncated cube}} = V_{\text{cube}} - V_{\text{removed}} = 27 - \frac{2\sqrt{3}}{3}
\]
**Final Answer**
The volume of the resulting polyhedron is:
\[
27 - \frac{2\sqrt{3}}{3}
\]
