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## Example Questions

### Example Question #19 : Solid Geometry

The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?

**Possible Answers:**

9√(3)

9

81

9√(2)

None of the others

**Correct answer:**

9√(3)

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x^{2}. This yields the equation:

6x^{2} = 486, which simplifies to: x^{2} = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x_{1} – x_{2})^{2} + (y_{1} – y_{2})^{2} + (z_{1} – z_{2})^{2})

For our data, this will be:

√( (–4.5 – 4.5)^{2} + (–4.5 – 4.5)^{2} + (4.5 + 4.5)^{2}) =

√( (–9)^{2} + (–9)^{2} + (9)^{2}) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

### Example Question #1 : How To Find The Diagonal Of A Cube

You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?

**Possible Answers:**

**Correct answer:**

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

### Example Question #2 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with side lengths of each?

**Possible Answers:**

**Correct answer:**

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

### Example Question #3 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube that has a surface area of ?

**Possible Answers:**

**Correct answer:**

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

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