3D geometry – Building a swimming pool (advanced)

Concepts addressed: percentages, unit conversion, cylinder volume, cylinder area, equilateral triangle area, regular hexagon area.
Recommended grade: 7th.
Difficulty level: Basic.
A basic version of Building a swimming pool is also available.


This is the year – you’re building an in-ground swimming pool in your backyard! You can already see all the envious faces, but first, someone’s got to do the dirty work.

  1. You’ve decided the pool should be circular, with a radius of six feet and as deep as you are tall. How much water will you need if you want to fill it up nine-tenths of the way (so that it doesn’t spill)? 

  1. You want to tile the entirety of the inside with regular hexagonal tiles that have a side 3 in long. How many of those will you need? Better buy 5% more in case some of them break in the process.

  1. A pack of thirty tiles costs $15, and you get $50 allowance every week. How long will it take to save for them if you already have $80 saved and you’ll be putting all your allowance towards it?

Useful calculators:

Question 1 hints:

Hint 1
What shape does the pool have in three dimensions? Can you describe its measurements?
Hint 2
How do we calculate the volume of a cylinder?

Question 2 hints:

Hint 1
How do we calculate the area we will have to tile? Remember that we don’t tile the top of our solid.
Hint 2
What is the area of a single tile? Recall that a regular hexagon can be divided into six equilateral triangles.
Hint 3
Knowing the area we have to tile, how many tiles do we need?
Hint 4
Remember that we need to have all values expressed in compatible units.
Hint 5
If we know how many tiles cover the area, how can we compute the additional 5%?

Question 3 hints:

Hint 1
How much money will be needed to buy all the necessary tiles? How much do you still have to save?
Hint 2
Knowing how much money we need, and that you get $50 allowance a week, how can we calculate the number of weeks that has to pass?

Solutions (WARNING: depend on the height, example for height being 6 feet ):

Question 1
V ≈ 610.4 ft³
Question 2
tiles = 2,226
Question 3
weeks = 9

Step-by-step solution:

Question 1
We want the pool to be circular, so the shape of it (and therefore the water inside too) will be cylindrical. From the scenario, we know that the radius of the base, r, is 6 ft. Also, if we are 6-feet tall, then this will be the pool’s depth, so h_cylinder = 6 ft. However, we don’t want to fill it all the way, but only nine-tenths of the way. Therefore, the height the water, h, will be:
h = h_cylinder * 9/10 = 6 ft * 9/10 = 5.4 ft.

This gives us all the information we need to calculate the volume with the cylinder volume calculator:
V = π * r² * h = π * (6 ft)² * 5.4 ft ≈ 610.4 ft³.

Question 2
The tiles will cover the bottom and the lateral face of our cylinder. We can think of the latter as a rectangle (whose two sides have been glued together) with one side equal to the cylinder height, h = 6 ft, and the other equal to the perimeter of the circle that is the base, b. We can use the circle calculator to find b:
b = 2 * π * r = 12π ft.
Now we can use that same calculator and get the area we want to tile:
A = base_area + lateral_area = π * r² + b * h = π * (6 ft)² + 12π ft * 6 ft 339.12 ft².

Now that we know how much area we’re working with, we need to get the number of tiles required to tile it. A single tile is a regular hexagon with side a = 3 in. For our calculations, we need the units to agree, so let’s use the conversion calculator to turn a into feet: a = 3 in = 0.25 ft.

Recall that a regular hexagon consists of six equilateral triangles, with sides equal to the hexagon’s side. Also, there is a nice formula for the area of an equilateral triangle that needs only the length of its side. In particular, this means that we are able to calculate the area of a single tile with the hexagon calculator:
A_tile = 6 * area_triangle = 6 * (a² * √3) / 4 = 6 * ((0.25 ft)² * √3) / 4 0.16 ft².
Therefore, we’ll need:
A / A_tile 339.12 ft² / 0.16 ft² = 2,119.5
tiles to cover the whole area. Clearly, you can’t buy half of a tile, so the actual number we need is 2,120.

Lastly, we need to add to that number the additional 5% in case some of the tiles break. If we want to buy 5% more than what is needed, then this means that we need 100% + 5% = 105% of what we calculated above. According to the percentage calculator this is 105% * 2,120 = 2,226 tiles.

Question 3
Let’s begin by calculating the cost of the tiles. From Question 2 we know that we need 2,226 of them. Now we must see how many packs of 30 tiles this translates to:
2,226 / 30 = 74.2.

Therefore, we’ll need 75 packs (since we cannot just buy 0.2 of a pack). This, in turn, boils down to
cost = 75 * pack_cost = 75 * $15 = $1,125.

Now let’s subtract from that the amount you’ve already saved to get how much we still need:
money_needed = cost – money_saved = $1,125 – $80 = $1,045.

Lastly, we must see how many allowances (and therefore weeks) it will take to save such a sum. Since we get $50 every week, we need 1,045 / 50 = 20.9, or rather 21 weeks, to get that much money. But summer will be over in 21 weeks!

Oh well, maybe next year will be the year?

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