The pool can be seen as a cylinder with depth (or height) , and a base with diameter  - and, subsequently, radius half this, or .

The swimming pool has one circular base, whose area is

The lateral side of the pool has area

Add these:

First, solve for circumference

The lateral surface area is equal to the circumference times the height. This gives us 

The lateral surface area of a right circular cylinder is equal to the circumference of the base multiplied by the height.

If the base is a circle with area , then the radius is , so

Therefore, circumference times height gives us

 .

Volume of a cylinder? V = πr²h. Rewritten as a diameter equation, this is:

V = π(d/2)²h = πd²h/4

Sub in h and V: 36p = πd²(4)/4 so 36p = πd²

Thus d = 6

LA = 2π(r)(h) = 2π(3)(5) = 30π

The equation for the volume of the cylinder is πr²h. When the radius doubles (r becomes 2r) you get π(2r)²h = 4πr²h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r. 

We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,

2πrh = 54π

Now we substitute 3r in for h.

2πr(3r) = 54π

6πr² = 54π

Divide by 6π

r² = 9.

Take the square root.

r = 3. 

h = 3r = 3(3) = 9.

Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr²h.

V = πr²h

V = π(3)²(9) = 81π

The answer is 81π.

The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πr_out² h – πr_in² h. The area of the cylinder using the outer radius is 80π cm³, and resulting hole is given by the volume from the inner radius, 20π cm³. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm³.

The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr².  Therefore, the total formula for the volume of the cylinder is: V = πr²h.

 First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.

Based on this, we know that the volume of our cylinder must be: π*12.5²*41.3 = 6453.125π in³

We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 2.5² * 8 = 50π in³

The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in³.