To make this problem easier to solve, we can inscribe a smaller square in the cube.  In the diagram above, points  are midpoints of the edges of the inscribed cube.  Therefore point , a vertex of the smaller cube, is also the center of the sphere.  Point  lies on the circumference of the sphere, so .   is also the hypotenuse of right triangle .  Similarly,  is the hypotenuse of right triangle .  If we let , then, by the properties of a right triangle, .

Using the Pythagorean Theorem, we can now solve for :

Since the side of the inscribed cube is , the volume is .

If the surface area is , then the area of one face must be . Therefore, the length of one edge must be  This means that the volume of the cube is . We can now solve with:

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

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 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. 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 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft²

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

For the door: 2.5 * 8 = 20 ft² 

For the windows: 3 * 6 = 18 ft²

The total wall space is therefore: 800 – 20 – 18 = 762 ft²

Now, if one can of paint covers 57 ft², we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.

Since we cannot buy partial cans, we must purchase 14 cans.

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

We have a cube with a surface area of 24, which means each side has an area of 4. Therefore, the length of each side is 2. If we double the height, the volume becomes .