ISEE Upper Level Quantitative : How to find the diagonal of a cube

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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

What is the length of the diagonal of a cube with a side length of  ? Round to the nearest hundreth.

Possible Answers:

 

 

 

 

 

Correct answer:

 

Explanation:

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions).  We could draw it like this:

Cube125

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

 and 

Solve this by using the distance formula.  This is very easy since one point is all s.  It is merely:

This is approximately .

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

What is the length of the diagonal of a cube with a side length of  ? Round to the nearest hundreth.

Possible Answers:

 

 

 

 

 

Correct answer:

 

Explanation:

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

Cube275

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

 and 

Solve this by using the distance formula.  This is very easy since one point is all s.  It is merely:

This is approximately .

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

What is the length of the diagonal of a cube with a volume of  ? Round to the nearest hundredth.

Possible Answers:

 

 

 

 

 

Correct answer:

 

Explanation:

First, you need to find the side length of this cube. We know that the volume is:

, where  is the side length.

Therefore, based on our data, we can say:

Solving for  by taking the cube-root of both sides, we get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

 

(Note that not all points are drawn in on the cube.)

Cube7

The two points we are looking at are:

 and 

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

Example Question #4 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with a surface area of  ? Round your answer to the nearest hundredth.

Possible Answers:

 

 

 

 

 

Correct answer:

 

Explanation:

First, you need to find the side length of this cube. We know that the surface area is defined by:

, where  is the side length. (This is because the cube is  sides of equal area).

Therefore, based on our data, we can say:

Take the square root of both sides and get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

 

Cube5

(Note that not all points are drawn in on the cube.)

 

The two points we are looking at are:

 and 

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

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