Bell Footing Volume for a Fence Post Hole
A straight cylinder is the right volume model for a fence-post hole only when the hole is straight. A bell footing is not straight. The shaft stays at one diameter, then the bottom flares to a wider diameter over a short height. Ordering concrete from the shaft diameter alone leaves that flare uncounted. Ordering the full depth as a cylinder at the wide diameter counts concrete the flare does not hold. The workable estimate splits the hole into a cylinder and a conical frustum, subtracts only the post that sits inside the concrete, and shows how a small change in diameter moves the bag count.
The arithmetic starts after the dimensions exist. It does not choose hole diameter, bell width, embedment, frost depth, reinforcement, concrete strength, or whether a bell belongs on the job. Those choices belong to the project detail, the fence system, the soil, and any local requirement. QUIKRETE's setting-posts page describes a straight hole, a gravel layer in the bottom, concrete fill, and soil backfill above the concrete. It also says that structural or load-bearing cases, such as deck-post footings, and posts in sandy soil, follow a different formed-footing procedure. It does not describe a bell. The QUIKRETE Concrete Mix 1101 data sheet lists approximate bag yields and tells load-bearing work to follow local building codes for footing specifications. Use those pages to keep gravel, concrete, and backfill on separate lines, and to read a bag yield. Do not read them as a bell design. A straight-hole worksheet is already in how to estimate concrete for fence post holes. Round forms that are not fence posts are in how to estimate concrete volume for round forms. After the net volume is in cubic feet, the concrete calculator can convert a uniform rectangular volume. It will not build the frustum.
Two shapes, and the formulas that match them
Measure the bell on a section sketch before multiplying anything. Record four lengths, all in the same unit. Shaft diameter is the constant width of the upper hole. Shaft concrete height is the vertical distance filled with concrete above the flare. Leave out gravel below the bell, and leave out any top backfill that is not concrete. Bell height is the vertical distance of the flare itself. Bell bottom diameter is the wide diameter at the base of the flare.
The model here is a right circular cylinder stacked on a conical frustum. The top of the frustum has the shaft radius. The bottom of the frustum has the bell radius. The sides of the frustum are straight in section. If the flare is a bulb, a step, or an undercut that a straight line does not describe, this formula does not apply. Split the hole into shapes that match the sketch instead of forcing a frustum onto a different solid.
Radius is half the diameter. Write both numbers on the sketch. Using the diameter in a radius formula multiplies the area by four, because the radius is squared. With radius and height in inches, shaft volume in cubic inches is pi × shaft radius² × shaft height. Frustum volume is bell = (pi × bell height / 3) × (R² + R × r + r²), where r is the shaft radius and R is the bell bottom radius. The term in parentheses is not the square of the average radius. An average-radius cylinder is a different formula. The two differ by (pi × bell height / 12) × (R − r)². The gap is small when the flare is slight and larger when the bell is much wider than the shaft. Use the frustum when the sketch is a straight-sided flare.
Gross concrete is the shaft plus the bell. The post removes volume only where the post and the concrete occupy the same space. If the post stops at the top of the bell, subtract post width × post thickness × shaft height and subtract nothing from the bell. If the post continues through the bell, extend the displacement through the bell height as well, using the post's actual cross-section. Do not subtract the above-ground portion of the post. A round post uses its own cylinder, not a square.
NIST Handbook 133 (2026), Appendix E, states that 1 cubic foot equals 1,728 cubic inches and that 27 cubic feet equal 1 cubic yard exactly (NIST Handbook 133, Appendix E). Divide cubic inches by 1,728 only after the net volume for the group of holes is totaled, then divide cubic feet by 27. Keep the unrounded cubic inches on the worksheet so a later reader can repeat the division. The inch-pound identities fix the conversion. They say nothing about how wide a footing should be.
Worked example: six matching bells
The numbers below are a hypothetical project detail used to show the arithmetic. They are not a recommended hole, not a manufacturer example, and not a field measurement.
Six line posts share one detail. Each shaft is 10 inches in diameter, so the shaft radius is 5 inches. The concrete shaft height is 24 inches. The bell is 6 inches high and 18 inches in diameter at the bottom, so the bell radius is 9 inches. Gravel, if the detail includes it, sits below the bell and is not inside these heights. Each post is an actual 3.5 inches by 3.5 inches and bears at the top of the bell, so it displaces shaft concrete only. Pi is the full-precision value 3.141592653589793. Displayed results are that arithmetic rounded to six decimal places. Recompute from the formula. Do not rebuild the total by adding the rounded pieces and then treating a one-unit difference in the last digit as a new measurement.
Shaft volume is pi × 5² × 24 = 1884.955592 cubic inches. Bell volume is (pi × 6 / 3) × (9² + 9×5 + 5²) = (pi × 2) × 151 = 948.760981 cubic inches. Gross volume is 2833.716574 cubic inches. Post displacement is 3.5 × 3.5 × 24 = 294 cubic inches exactly. Net volume per hole is 2539.716574 cubic inches, which is 2539.716574 ÷ 1728 = 1.469743 cubic feet from the full-precision net, not from the already-rounded cubic-inch display.
Six matching holes are 8.818460 cubic feet, or 8.818460 ÷ 27 = 0.326610 cubic yards. Multiply the full-precision net by six before rounding. Multiplying the displayed 1.469743 by six and then dividing can drift in the last digit.
The same six holes with no bell, shaft only, are 0.204598 cubic yards. The bells contribute 0.122011 cubic yards. Those two rounded parts sum to 0.326609, which is 0.000001 cubic yard under the full-precision total of 0.326610. The difference is rounding, not missing concrete. For this detail the flare is still more than a third of the order. Leaving it out is not a rounding choice.
A unit cross-check in feet should land on the same net. Shaft radius 5/12 foot and height 2 feet give 1.090831 cubic feet. The bell, with height 0.5 foot and radii 5/12 and 9/12 foot, gives 0.549051 cubic feet. The post, 3.5/12 foot square through 2 feet, displaces 0.170139 cubic feet. The net is 1.090831 + 0.549051 − 0.170139 = 1.469743 cubic feet, the same six-decimal result as the inch worksheet. If the two unit systems disagree past the last displayed digit, one of them mixed inches and feet inside a single product.
Why a full cylinder and an average radius both miss
Two shortcuts show up on sketches, and both are the wrong solid for a straight-sided flare.
The full-cylinder shortcut treats the bell as an 18-inch-diameter cylinder for the whole 6-inch height. That volume is pi × 9² × 6 = 1526.814030 cubic inches, which is 578.053048 cubic inches above the frustum. Per hole that is 0.334521 cubic feet. Across six holes it is 2.007129 cubic feet, or 0.074338 cubic yards. At the 80 lb yield used below, 2.007129 ÷ 0.60 = 3.345215 bags. Those bags are not in the hole. They are an artifact of ignoring the taper. The shortcut is the right model only when the lower portion really is a cylinder, with a flat shelf rather than a slope. That is a stepped footing, and it should be labeled as two cylinders.
The average-radius shortcut uses radius (9 + 5) / 2 = 7 inches over the bell height: pi × 7² × 6 = 923.628240 cubic inches. That is 25.132741 cubic inches, or 0.014544 cubic feet, below the frustum on each hole, and 0.087266 cubic feet across six holes. The gap equals (pi × 6 / 12) × (9 − 5)², which is 25.132741 cubic inches, the identity above. On this mild flare the bag count may not move. The worksheet is still no longer the volume of the sketch. When the bell is wider relative to the shaft, the same identity grows with the square of (R − r). A 4-inch difference in radius is already visible. A 6-inch difference is more than twice the volume error, because the square goes from 16 to 36. Keep the frustum unless a different shape is written on the section.
A bracket for diameter tolerance
Each diameter in the example is treated as exact. A field hole is not. Suppose each diameter could be off by 0.25 inch in either direction, and the heights and the post stay fixed. The low case uses a shaft radius of 4.875 inches and a bell radius of 8.875 inches. The high case uses 5.125 and 9.125. Recompute both volumes from the same two formulas. Do not scale the point estimate by a flat percentage. Radius is squared, so a quarter-inch of diameter does not move volume by a quarter-inch's share of the diameter.
The low net for six holes is 8.381787 cubic feet, or 0.310437 cubic yards. The high net is 9.265360 cubic feet, or 0.343161 cubic yards. The bracket covers only that diameter tolerance. It does not cover a hole that caves in after you measure, concrete left in a wheelbarrow, or a post that is not the size on the sketch. If you add an allowance for those, put it on its own line with a reason. Do not fold it into the frustum, and do not widen the diameter bracket to hide a different uncertainty.
Read the bracket against the purchase unit, not against the cubic-yard display alone. A quarter-inch that looks small next to an 18-inch bell can still cross a bag boundary once six holes are added. The next section does that comparison with one product's published yields.
From cubic feet to bags
Bag count uses the yield printed for the product you are buying, not the weight printed on the bag. Weight and volume are different quantities. The QUIKRETE Concrete Mix 1101 data sheet lists an 80 lb bag as yielding approximately 0.60 cubic feet and a 60 lb bag as approximately 0.45 cubic feet. Those figures are approximate, they depend on mixing water and consolidation, and they belong to that product. The sheet was revised October 2022 and tells readers to check the current technical data. Read the bag in front of you before you divide.
Point estimate, 80 lb bags: 8.818460 ÷ 0.60 = 14.697434, which rounds up to 15 bags if the seller sells whole bags. Point estimate, 60 lb bags: 8.818460 ÷ 0.45 = 19.596578, which rounds up to 20 bags. Diameter bracket, 80 lb bags: 8.381787 ÷ 0.60 = 13.969645 to 9.265360 ÷ 0.60 = 15.442267, or 14 to 16 bags after rounding each end up to a whole bag.
The bracket surrounds the point estimate rather than sitting neatly on it. A single line that says 15 bags hides that a quarter-inch of diameter, applied the same way to every hole, can move the purchase by one bag on either side. If the holes are not actually identical, do not multiply one hole by six. A gate post or a corner post with a larger bell is a second group, with its own shaft, its own frustum, and its own count.
For a quantity this small, bags are the usual purchase. If a supplier sells by the cubic yard and imposes a minimum, compare 0.326610 cubic yards with that minimum before converting to bags. Concrete bags versus ready-mix walks through that comparison for larger volumes. The supplier's increment, short-load fee, and rounding rule belong to the supplier. They are not a result of the geometry.
Failure modes
Diameter used as radius. A 10-inch shaft entered as a 10-inch radius produces pi × 10² × 24 = 7539.822369 cubic inches of shaft, four times the 5-inch-radius shaft of 1884.955592 cubic inches. The bell fails the same way if 18 is used as a radius. Put diameter and radius in separate columns, and compute the radius on the sheet rather than in your head.
Gravel counted as concrete. The setting-posts page puts gravel in the bottom of a straight hole and concrete above it. If your detail does the same under a bell, the gravel height is not bell height. Adding it to the frustum buys concrete for a space the gravel occupies. The opposite error is subtracting gravel and also shortening the bell by the same distance. Subtract a non-concrete layer once, on the section, before either formula runs.
Post subtracted through the wrong height. Subtract the post only inside the concrete. A post that stops on top of the bell does not reduce the bell. A post subtracted for the full embedment, including a length sitting in gravel, understates the concrete. Subtracting the post from the gross and then subtracting it again as a waste factor counts the same wood twice.
One average hole for a mixed run. Corner posts, gate posts, and line posts often have different details. Six times one bell is valid only when six holes share the sketch. Keep a count column next to each detail so a post that was added to the fence layout is not still missing from the concrete.
Nominal lumber as the post size. A label such as 4×4 is not a measurement of 4 inches by 4 inches. This example uses 3.5 inches because that was the stated actual size. Displacement is the smaller term, but it should still be the piece you are setting. Measure the actual section, or copy it from the product specification, and write which one you used.
A frustum on a hole that is not a frustum. An undercut, a hand-dug bulge, or a stepped footing is a different solid. A stepped footing is two cylinders: the shaft, plus a full-width lower cylinder. Using the frustum on a step understates the lower cylinder, which is the full-width volume both the average-radius formula and the frustum reduce on purpose. Match the formula to the section you can point at. If the walls are not straight, measure a second diameter at mid-height of the flare and stop claiming a single frustum.
Limits, and what to keep with the order
This worksheet estimates the volume of a stated cylinder plus a stated frustum, minus a stated post. It does not check whether the footing can resist wind, frost heave, or uplift, and it does not set concrete strength, reinforcement, cover, drainage, or cure time. The cited QUIKRETE pages do not design a bell. The setting-posts page sends structural footings and sandy soil to a formed procedure, and the 1101 sheet tells load-bearing work to follow local building codes for footing specifications. NIST Appendix E supplies the cubic-inch and cubic-yard identities only.
Keep the section sketch, the four lengths, the post size, the count of holes that share the detail, the value of pi you used, the bag yield and the product name, and the date. A reviewer should be able to recompute 0.326610 cubic yards, and the 14-to-16 bag bracket, without guessing which diameter was the radius. If a dimension on the sketch changes, replace the row. Do not average the old bell with the new one.
Sources and method
- NIST Handbook 133 (2026), Appendix E: General Tables of Units of Measurement — retrieved 2026-10-23
- QUIKRETE Concrete Mix Product No. 1101 Data Sheet — retrieved 2026-10-23
- QUIKRETE: Setting Posts in Concrete — retrieved 2026-10-23
AI-assisted writing with automated source, calculation and editorial checks. No hands-on testing or professional certification is claimed. Check the stated assumptions before applying this guide.