A thin rod of length 24 feet is cut into rods of equal size and joined so as to form a skeleton cube. What is the area of one of the faces of the largest cube thus constructed?
4 square feet
Calculating Cube Face Area from Rod Length
The question asks us to determine the area of one face of the largest skeleton cube that can be formed by cutting a thin rod of length 24 feet into equal pieces.
A skeleton cube is made up of 12 edges. For a cube, all these edges are of equal length. The total length of the rod is used to form these 12 edges.
To find the length of each edge of the cube, we need to divide the total length of the rod by the number of edges in a cube.
- Total length of the rod = 24 feet
- Number of edges in a cube = 12
Length of one edge of the cube (let's call it 's') can be calculated as:
\(\text{s} = \frac{\text{Total rod length}}{\text{Number of edges}}\)
\(\text{s} = \frac{24 \text{ feet}}{12}\)
\(\text{s} = 2 \text{ feet}\)
So, each edge of the constructed cube is 2 feet long.
The area of one face of a cube is the area of a square with side length equal to the edge length of the cube. The formula for the area of a square is side × side, or side squared (\(\text{side}^2\)).
Area of one face = \((\text{edge length})^2\)
Area of one face = \((2 \text{ feet})^2\)
Area of one face = \(4 \text{ square feet}\)
Therefore, the area of one face of the largest cube constructed from the 24-foot rod is 4 square feet.
Step-by-Step Solution for Cube Area
- Identify the total length of the material (rod) available: 24 feet.
- Recognize that a skeleton cube consists of 12 equal edges.
- Calculate the length of one edge by dividing the total rod length by the number of edges: \(24 \text{ feet} / 12 = 2 \text{ feet}\).
- Understand that a face of a cube is a square with side length equal to the cube's edge length.
- Calculate the area of one face using the formula for the area of a square: \((\text{edge length})^2 = (2 \text{ feet})^2 = 4 \text{ square feet}\).
| Parameter | Value |
|---|---|
| Total Rod Length | 24 feet |
| Number of Edges in Skeleton Cube | 12 |
| Length of One Edge | \(\frac{24}{12} = 2\) feet |
| Shape of Cube Face | Square |
| Side Length of Face | 2 feet |
| Area of One Face | \((2 \text{ feet})^2 = 4\) square feet |
Revision Table: Key Geometric Concepts
| Concept | Definition/Formula | Relevance to Problem |
|---|---|---|
| Cube | A 3D shape with 6 square faces, 12 edges, and 8 vertices. All edges are equal in length. | The problem is about forming a skeleton cube. |
| Skeleton Cube | Represents only the edges of the cube. Formed by 12 edges. | The rod forms the edges of this structure. |
| Edge Length | The length of one side of the cube. All 12 edges have the same length. | Calculated by dividing total rod length by 12. |
| Area of a Square | Side length × Side length (or side length squared). | Used to find the area of one face of the cube (which is a square). |
Additional Information on Cubes and Area
A cube is a fundamental shape in geometry. Understanding its properties is key to solving problems like this one. Here are some additional points about cubes and area calculations:
- The total surface area of a cube is the sum of the areas of its 6 faces. If the edge length is 's', the total surface area is \(6s^2\).
- The volume of a cube with edge length 's' is \(s^3\).
- Area is always measured in square units (like square feet, square meters, square inches), because it represents a two-dimensional space. Length is measured in linear units (like feet, meters, inches).
- This problem involves converting a total linear length into the dimensions of a 3D shape (specifically, its edges) and then calculating a 2D property (area of a face).
- The term "largest cube thus constructed" implies that all of the rod's length is used efficiently to make the longest possible edges, resulting in the largest possible cube from the given material.
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