The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √( (x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or for our simpler case:

d = √( (x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (8)² + (8)²  + (8)²) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 1728, or (taking the cube root of both sides), s = 12.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):

d = √( (x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or, for our simpler case:

d = √( (x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (12)² + (12)²  + (12)²) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)

 The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6s², where s is the length of the cube. For our data, this is:

6s² = 294

s² = 49

(taking the square root of both sides) s = 7

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √((x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or for our simpler case:

d = √((x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (7)² + (7)²  + (7)²) = √( 49 + 49 + 49) =  √(49 * 3) = 7√(3)

The volume of a rectangular prism is .

We are told that the shortest edge is 3.  Let us call this the height. 

We now have , or .

Now we replace variables by known values:

Now we have:

We have thus determined that the other two edges of the rectangular prism will be 4 and 12.  We now need to find the longest diagonal.  This is equal to:

If you do not remember how to find this directly, you can also do it in steps.  You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem.  For example, we choose the side with edges 3 and 4.  This diagonal will be:

We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12). 

This diagonal is then .

This problem is relatively simple. We know that the volume of a cube is equal to s³, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s³ = 1728. Taking the cubed root, we get s = 12.

Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in². Therefore, the total surface area is 6 * 144 = 864 in².

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft²

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 3 * 7 = 21 ft² 

For the windows: 2 * (2 * 5) = 20 ft²

The total wall space is therefore: 594 – 21 – 20 = 553 ft²

A cube with a side length of 25m has a surface area of:

25m * 25m * 6 = 3,750 m²

(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6s^2.)

Each square tile has an area of 5 m².

Therefore, the total number of square tiles needed to fully cover the surface of the cube is:

3,750m²/5m² = 750

Note: the volume of a cube with side length s is equal to s³.  Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:

s³/n³

To begin, we need to solve for the dimensions of the cone.

The basic form for the volume of a cone is: V = (1/3)πr²h

Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).

486π  = (1/3)πr² * 2r = (2/3)πr³

Multiply both sides by (3/2π): 729 = r³

Take the cube root of both sides: r = 9

Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.

The surface area of the cube is defined by: A = 6 * s², or for our data, A = 6 * 111² = 73,926 in²

The volume of a cube is given by the formula V = , where V is the volume, and s is the length of each side. We can set V to 216 and then solve for s.

In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.

This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to . The formula for surface area comes from the fact that each face of the cube has an area of , and there are 6 faces in a cube.

surface area =

The surface area of the square is 216 square units.

The answer is 216.

The area of one side of the cube is:

A cube has 6 sides, so the total surface area of the cube is