Cubes: True / false

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

• Step 1: Identify the properties of a cube.
• Step 2: Count the number of faces and relate to surface area.

Let's go through each step:
Step 1: A cube is a three-dimensional shape with all sides equal in length and each angle a right angle. A cube has 6 faces, each of which is a square.
Step 2: The surface area ($A$) of a cube is calculated as $A = 6s^2$, where $s$ is the length of a side of the cube. The calculation considers contributions from all 6 faces, each being square, hence a cube having 6 faces is integral to the computation of its surface area. The number of faces is 6 and each is involved in computing the surface area through this formula.

Therefore, the statement that all cubes have 6 faces equating to the surface area property is Yes..

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.

To determine whether the volume of a cube is equal to the length of the edges cubed, we follow these steps:

• Step 1: Recognize that a cube is a three-dimensional shape with six equal square faces. All edges of a cube are of equal length.
• Step 2: Recall the formula for the volume of a cube, given by $V = s^3$, where $s$ is the length of one of the cube's edges.
• Step 3: The question asks if the volume is equal to the edge length cubed. We note that the formula $V = s^3$ clearly indicates that the volume of a cube is indeed calculated by cubing the length of its edges.

Thus, the volume of a cube is $\mathbf{equal}$ to the length of the edges cubed.

Therefore, the correct answer to this problem is:

Yes

To solve this problem, we need to understand the definitions and properties of cuboids and cubes:

• A cuboid is a three-dimensional shape with six faces, all of which are rectangles. Typically, the lengths of the edges can differ from each other.
• A cube, however, is a special type of cuboid where all faces are squares and all three dimensions (length, width, height) are equal.

Given these definitions:

• Every cube is a cuboid because it satisfies the requirement of having six rectangular faces (which are squares).
• However, not every cuboid is a cube because a cuboid does not need to have equal side lengths. In many cases, cuboids have different side lengths, which disqualifies them from being cubes.

Therefore, we can conclude that the statement "every cuboid is a cube" is false. There are many cuboids that are not cubes because they lack the property of equal side lengths.

Thus, the correct answer to the problem, "Is every cuboid a cube?" is no.

To determine if each cube is a cuboid, we start by defining both shapes.

• A cuboid is a three-dimensional geometric figure with six faces that are rectangles. All angles are right angles, and opposite faces are equal.
• A cube is a special type of cuboid where all six faces are squares of equal size, meaning all sides (edges) of the cube are of equal length.

Since a cube meets all the criteria of a cuboid (having six rectangular, or in this case square, faces, with all angles being right angles), a cube can indeed be classified as a cuboid.
In mathematical terms, a cube is a specific case of a cuboid where the length, width, and height are all the same.

Therefore, the answer to the question “Is each cube a cuboid?” is Yes.

To solve this problem, we'll clarify the definitions and properties of a cuboid:

• A cuboid is a three-dimensional geometric figure with six rectangular faces, where opposite faces are equal and parallel. Importantly, a cuboid has dimensions that are specified as length, width, and height, and these values can be different from one another.
• A cube is a special type of cuboid where all three dimensions (length, width, and height) are equal. This makes all faces of a cube identical squares.

Given these definitions, we analyze the problem:

Since a cuboid's dimensions can be different, it follows that the height of a cuboid can indeed be different from its length. This contrasts with a cube where all dimensions must be equal.

Hence, we conclude that it is possible for a cuboid to have a height that is different from its length.

Therefore, the answer to the question is Yes.

To determine whether a cube can have a height that is different from its length, we must recall the definition of a cube.

A cube is a special type of three-dimensional geometric shape wherein all the sides are equal in length. In mathematical terms, if a cube has a side length of $a$, then its dimensions are $a \times a \times a$. This means the length, width, and height are identical.

Since the height and length of a cube are both represented by the same variable, since their definitions require that they are exactly the same length, a cube cannot have a height that is different from its length.

Therefore, according to the geometric properties that define a cube, the height cannot be different from its length.

Thus, the answer to the question "Can a cube have a height that is different to its length?" is No.

To solve this problem, we will follow these steps:

• Step 1: Establish the geometric relationship between the edge and diagonal.
• Step 2: Use algebra to explore if equality is possible.

Now, let's proceed through each step:

Step 1: For a cube with edge length $a$, the face is a square with sides $a$, and its diagonal $d$ can be found using the Pythagorean theorem:

$d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}$

Step 2: We consider if it is possible for the edge length to equal the diagonal. Thus, we set

$a = a\sqrt{2}$

Dividing both sides by $a$, assuming $a \neq 0$, we get:

$1 = \sqrt{2}$

However, this is not correct since $\sqrt{2} \approx 1.414$ is not equal to $1$. This shows that it is impossible for the edge length and the diagonal of the face to be equal.

Therefore, the solution to the problem is No.