A plane can be named after any three points on the plane that are not on the same line. As seen below, points , ,  and  are on the same line. 

Therefore, Plane  is not a valid name for the plane.

Two rays are opposite rays, by definition, if 

(1) they have the same endpoint, and

(2) their union is a line.

The first letter in the name of a ray refers to its endpoint; the second refers to the name of any other point on the ray.  and  both have endpoint , so the first criterion is met.  passes through point  and  passes through point ;  and  are indicated below in green and red, respectively:

The union of the two rays is a line. Both criteria are met, so the rays are indeed opposite.

A line can be named after any two points it passes through. The line  is indicated in green below.

The line does not pass through , so  cannot be part of the name of the line. Specifically,  is not a valid name.

By definition, two angles comprise a pair of vertical angles if 

(1) they have the same vertex; and

(2) the union of the two angles is exactly a pair of intersecting lines.

In the figure below,  and  are marked in green and red, respectively:

While the two angles have the same vertex, their union is not a pair of intersecting lines. The two angles are not a vertical pair.

By definition, two angles form a linear pair if and only if 

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

In the figure below,  and  are marked in green and red, respectively:

The two angles have the same vertex and share no interior points. However, they do not share a side. Therefore, they do not comprise a linear pair.

To solve this, we need to break it down into three parts. 

The number of square units in the surface area of a cube is given by the formula 6s², where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s³. 

Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:

6s² = 2s³

Subtract 6s² from both sides.

2s³ – 6s² = 0

Factor out 2s² from both terms.

2s²(s – 3) = 0

We must set each factor equal to zero. 

2s² = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.

Set the other factor, s – 3, equal to zero.

s – 3 = 0

Add three to both sides.

s = 3

This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s³, must then be 3³, or 27 cubic units.

The answer is 27. 

You own a Rubik's cube with a volume of . What is the edge length of the cube?

To solve for edge length, think of the volume of a cube formula:

Now, we have the volume, so just rearrange it to solve for side length:

General formula for the diagonal of a cube if each side of the cube = s

Use Pythagorean Theorem to get the diagonal across the base:

s² + s² = h²

And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:

h² + s² = d²

s² + s² + s² = d²

3 * s² = d²

d = √ (3 * s²) = s √3

So, if s = 3 then the answer is 3√3

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere.  Since the radius = 1, the diameter = 2.