The Pool Dose Figured from the Diameter as the Radius

Why this matters

Every chemical dose on a pool scales with the gallons. Get the gallons four times too high and every product goes in at four times the label dose, and the first chemical where that hurts is acid. On an 18 ft round above-ground pool, a tech who used the diameter where the formula wanted the radius dosed for about 30,000 gallons into about 7,600, and the pH went off the bottom of the test kit within the hour.

This case runs in the order it had to: get people out of the water and stop the damage, then bring the water back, then work out how the number went wrong. The arithmetic took ten seconds to fix. The water took two days.

What the note said

The tech's dosing note from the visit, reconstructed from the service ticket:

POOL: 18 ft round above-ground, vinyl liner
WATER DEPTH: 4 ft
TEST: pH 7.9   alkalinity 120 ppm
TARGET: pH 7.4 to 7.6

GALLONS  3.14 x 18 x 18 x 4 x 7.48  = 30,440 gal
ACID     label dose D per 10,000 gal for this drop
         30,440 / 10,000 = 3.04, so 3 x D
ADDED    3 x D muriatic acid at return, pump on

SHOULD HAVE BEEN
GALLONS  3.14 x 9 x 9 x 4 x 7.48    = 7,610 gal
ACID     7,610 / 10,000 = 0.76, so 3/4 x D

The pool got 3 x D where it needed about 0.76 x D. Three divided by 0.76 is 3.9 times the dose: the gallons were four times too high, and rounding 3.04 down to 3 took a sliver off.

Stop the damage first

The tech retested before leaving, as the shop's standard requires after an acid addition, and the phenol red reagent went to its yellowest color, the bottom of the scale on a standard kit, about pH 6.8. The real pH was somewhere below that, and nobody on site could say how far. Alkalinity had also fallen sharply, which removes the water's buffer and lets the pH swing on the next small addition.

The order of the next fifteen minutes:

Swimmers out and kept out. Water well below 7 stings eyes and skin and is not a pool anyone uses until it reads in range. The tech told the homeowner in person and taped the gate with a note, because a verbal warning does not reach a neighbor's kid on a hot afternoon.

Heater off, if the pool has one. Low pH water attacks metal, and a heat exchanger is the most expensive metal the water touches. This pool had no heater. Where one is fitted, turn it off at its control and leave the pump running, so the water keeps moving without passing hot over the exchanger. It goes back on only once pH and alkalinity read in range, and the tech confirms it fires and heats normally.

Pump running, nothing else added. Circulation mixes the acid through the full volume, so a retest after the water turns over reads the pool rather than one corner of it. What the tech did not do is reach for the pH increaser immediately: a second dose figured off a gallon number nobody had yet questioned would have been a second guess.

Protect yourself while handling it. Muriatic acid gives off a vapor that attacks the airways and eyes; pour it with eye protection and acid-resistant gloves, standing upwind, low to the water. Never add water to acid, and never let acid meet a chlorine product, in the bucket, the tray or the truck, because together they release chlorine gas. If acid gets on skin or in an eye, flush with running water for at least 15 minutes and get medical attention for an eye.

Find the number that was wrong

The tech called the office from the driveway. The service manager asked one question first: "What size is the pool?" Eighteen-foot round, four feet of water. "Then it's around seven and a half thousand gallons. Where did thirty come from?"

That question is the whole diagnosis, and it works because it uses a number nobody calculated on site. An 18 ft round pool with 4 ft of water is a small above-ground pool. Thirty thousand gallons is a large in-ground pool, something like 20 by 40 ft at 5 ft average depth (20 x 40 x 5 = 4,000 cu ft, x 7.48 = 29,920 gal). The two pools are not close in size, and anyone who has stood next to both knows it.

The note shows where it went wrong. The area of a circle is pi times the radius squared. The radius of an 18 ft circle is 9 ft. The note squared 18:

• Wrong: 3.1416 x 18 ft x 18 ft = 1,017.9 sq ft.
• Right: 3.1416 x 9 ft x 9 ft = 254.5 sq ft.

Squaring a number twice as big gives a result four times as big, so every downstream figure was four times too high: 1,017.9 / 254.5 = 4.0. The full volume chain, corrected:

• 254.5 sq ft x 4 ft = 1,017.9 cu ft.
• 1,017.9 cu ft x 7.48 gal per cu ft = 7,614 gal.

The 7.48 comes from The Water Constants 7.48, 8.34 and 231, and the radius rule from Circles in the Field: Area, Circumference and the Diameter Trap. The tech knew both. The slip was writing down the diameter, which is the number you measure and the number the customer gives you, in the slot that wanted half of it.

Bring the water back

With the volume right, the correction can be figured honestly. The goal is to restore alkalinity first, because it is the buffer that lets the pH hold, and then let the pH settle into range.

Dose on the corrected gallons, and in parts. The tech used sodium bicarbonate, which raises alkalinity with a smaller effect on pH than soda ash, and figured it from the product label's chart at 7,614 gallons. Then he added half of the calculated amount, broadcast across the surface with the pump running, holding back the rest. Bicarbonate is a dust; pour it low and keep your face out of the cloud.

Retest after the water has turned over. On this pool that meant coming back the next morning. Alkalinity had come up and pH read on the scale again, low in the range. The second half of the bicarbonate went in, and the third visit read pH 7.4 and alkalinity in the shop's target band.

Do not drain to dilute without checking the liner. Partial draining and refilling is a real way to correct water chemistry, but lowering the water in a vinyl-lined above-ground pool can let the liner shift or shrink, and the liner and pool manufacturer's instructions set how low it can go. Here the chemical correction was enough.

Look at what the water touched. With the pH restored, the tech checked the ladder hardware, the return fitting, the skimmer and the pump and filter for corrosion, staining or a softened seal, and looked over the liner for new wrinkling. Nothing showed after the one day at low pH, and the ticket says so in writing in case it does later.

The second slip that almost happened

The first correction the tech worked out on the phone used 30,440 gallons again, out of habit, because it was the number written on the note. The service manager caught it only because she had already said "seven and a half thousand" out loud. A wrong volume on a ticket does not stay in one calculation; it becomes the input to every dose after it, including the one meant to fix the damage. The note now gets the wrong figure struck through and the corrected one written beside it, rather than left for the next person to copy.

The same slip running the other way

The mirror error is quieter and in one way worse. A tech who has learned the 5.9 shortcut but feeds it the radius gets 5.9 x 9 x 9 x 4 = 1,912 gal, a quarter of the real volume. Every dose figured from that is a quarter of what the pool needs.

On acid that means a pH that barely moves, and the tech adds more on the next visit. On chlorine it means the sanitizer level never reaches the label's range, the water looks fine for a day or two, and then it does not. An underdose produces no alarm at the test kit the way the acid overdose did; it shows up as a cloudy or green pool and a callback, with the cause sitting in a gallon figure nobody rechecks because it never caused a crash.

That is why the fix is the size check rather than the formula. Both formulas are correct and both fail when they get the wrong half of the circle. The check against the pool's known size catches 30,440 and 1,912 equally, because both are wildly off from a small above-ground pool's 7,600.

What changed afterward

The 5.9 shortcut for round pools. For a round pool, gallons = 5.9 x diameter x diameter x average depth, all in feet. The constant already contains the quarter that turns a diameter into a radius (pi / 4 x 7.48 = 5.87, rounded to 5.9), so the tech can use the number he measured and never halve anything. For this pool: 5.9 x 18 x 18 x 4 = 7,646 gal, within half a percent of the exact 7,614. It takes the number the tech actually measured, so there is no halving to forget; it can still be fed a radius, as the mirror case shows, which is why the size check below stays. See related: How to Estimate Pool and Spa Gallons.

A sanity check against the pool's known size before any acid goes in. Gallons roughly equal 7.5 x surface area x average depth, and a tech who knows the pool can tell a small above-ground from a large in-ground at a glance. The dosing note now carries a line asking for that judgment in words: "small above-ground, about 7,600, check".

Gallons on the customer record, computed once. The pool's volume is worked out once, by someone with time to check it, and saved on the customer record with the dimensions it came from. Every dose after that reads the stored figure. A stored figure can be wrong too, but it is wrong once, in a place someone can audit, instead of recomputed on a hot driveway every visit.

The retest stays mandatory. The rule that caught this, retest after every acid addition before leaving, is the rule that kept a bad dose from becoming a damaged liner and a customer swimming in it overnight. Nothing in the fix loosened it.

References

• The chemical manufacturer's label and dosing chart for each product used
• The pool and liner manufacturer's instructions on water level and chemistry ranges
• See related: Circles in the Field: Area, Circumference and the Diameter Trap
• See related: How to Estimate Pool and Spa Gallons
• See related: The Water Constants 7.48, 8.34 and 231