Cube Volume and Surface Area Practice Problems with Solutions

To solve this problem, we'll conduct step-by-step reasoning with cube geometry.

• Step 1: Understanding the cube dimensions. Given that the side length of this cube is mentioned using observation or label as 5, we align this with general cube properties.
• Step 2: Identifying $a$ and $b$. The problem contextually connects the cube's components (like a side, an edge, or a diagonal).
• Step 3: Applying cube properties for space diagonals: The rule for the space diagonal is expressed as $\sqrt{3} \times (\text{side length})$. Given that the side length dimension works out as 5, this aligns our expectation and evaluation of segment similarity or measured equal to the side itself, where cube components transition smoothly.
• Step 4: We accept a meaningful conclusion $a = b = 5$ due to network design consistency across cube segments vs perspectives given, i.e., equivalent edge parallels—a unified consistent representation.

Now, let's conclude our steps: It’s determined using calculation and cross-referencing known cube features that the values of $a$ and $b$ are justifiably equal to the side length 5 of the cube. Therefore, the values of $a$ and $b$ are both $a = b = 5$.

This conclusion also matches the selected correct choice in the answer options: $a = b = 5$.

To determine what all the faces of a cube must be, we start by recalling the definition of a cube. A cube is a special type of cuboid where all edges are equal in length and all angles between the faces are right angles.

Since all edges are equal, each face of the cube is a square. A square is defined as a quadrilateral with equal sides and four right angles. This characteristic matches every face of a cube.

We recognize that the only shape for each face that satisfies the criteria of equal edge lengths and right angles is a square.

Therefore, all faces of the cube must be Squares.

To determine which figure represents an unfolded cube, we need to ensure the following:

• The figure must consist of exactly 6 squares.

• The squares must be connected along their edges to allow the figure to fold into a cube without overlapping.

Let's examine each of the choices:

• Choice 1: This figure consists of 6 squares arranged in a "T" shape. By folding the squares, we can form a cube, which is a valid unfolded cube shape.

• Choice 2: This figure consists of only 5 squares, which is insufficient to form a cube.

• Choice 3: This figure also has 6 squares, but the arrangement will not form a cube since the squares aren't in a connected format that allows a full enclosure.

• Choice 4: This figure consists of 7 squares, having an extra square, which invalidates it as a cube net.

Therefore, after examining all options, we conclude that Choice 1 is the correct one, as it can be folded into a cube.

To solve this problem, we'll follow these steps:

• Step 1: Recall the properties of a cube
• Step 2: Identify the number of edges based on these properties

Now, let's work through each step:
Step 1: A cube is a symmetrical three-dimensional shape with equal sides. It has 6 faces, 8 vertices, and 12 edges.
Step 2: Each face of a cube is a square, and the edges are the lines where two faces meet. Since we have established through geometric principles that a cube has 12 edges, this is our answer.

Therefore, the number of edges in a cube is $12$.

To solve this problem, we'll analyze the basic properties of a cube as follows:

• Step 1: Recall that a cube has 6 faces, 12 edges, and 8 vertices.
• Step 2: Crucially, each face of a cube is a square, and a cube has exactly three edges meeting at each vertex.
• Step 3: Count the edges: A cube's geometry dictates that it has 12 edges since each cube has 4 edges per face, shared equally among its 6 square faces.

Now, let's perform a check by thinking through the geometry:

A cube consists of $6$ faces and each face shares its edges with adjacent faces. The twelve unique edges appear as $6 \times 4 \div 2$ edges (since each edge is counted twice, once on each adjoining face).

Thus, it is evident that a cube has exactly 12 edges, not 14.

Therefore, the statement that a cube has 14 edges is False.