Apparatus Competition

2007 AAPT Summer Meeting

Greensburo, NC


Scaling With Bolts


Brett Carroll

Green River Community College

12401 SE 320th St.

Auburn, WA 98092


253-833-9111  x4322



When an object is “scaled up” by a factor X, its surface area increases by X2 and its volume increases by X3.  Students often struggle with this concept, particularly regarding objects with irregular shapes.  Stacking cubes into larger cubes is helpful, but does little to eliminate questions about “real world” scaling.


After much searching, industrial bolts have proven an excellent example of scaling of irregular objects.  They are available in many sizes in ratios of 1:2:3:4, and lend themselves easily to investigations of surface area and volume ratios.

Construction of Apparatus: 

In its simplest form, scaling with bolts can be explored with nothing more than a set of bolts, some paper, and a scale or graduated cylinder to measure volume. 


Standard bolts are specified using two sets of numbers – the thread size, a pair of numbers that indicate both the diameter and number of threads per inch of the bolt, and the length of the threaded portion.  A typical designation is ¼”-20 by ½”, indicating a bolt with a ¼ inch diameter and 20 threads per inch that is ½ inch long.

The following is a very useful set of bolts for scaling investigations, and can be found at McMaster-Carr (


¼”-28 by ½”         (McMaster-Carr #92620A562, 27 or more)

½”-13 by 1”          (McMaster-Carr #92620A712, 8 or more)

¾”-10 by 1 ½”      (McMaster Carr #92620A839, 3 or more)

1”-8 by 2”             (McMaster-Carr #92620A953, 1 or more)


These are in linear ratios of 1:2:3:4, simplifying the mathematics required to compare measurements.

Area comparisons can be done with paper and pencil, or simply by stacking the bolts to see how many heads of a smaller bolt it takes to cover the head of a larger bolt etc.

Volume comparisons can be done most simply be weighing the bolts to see how many of a smaller bolt it takes to equal the weight of a larger bolt.  Since weight is proportional to the amount of steel in a bolt and thus to its volume, this gives a very accurate comparison of volumes.  Another way to measure volumes is through water displacement in a graduated cylinder or other tall thin transparent container.

Use of Apparatus: 

Although scaling of surface area and volume a objects get larger are important concepts for the understanding of many physical systems, good real-world examples that can be analyzed easily are difficult to find.  To work well, the objects must be available in a variety of sizes in simple ratios, have a somewhat complex (but not too complex!) geometry, be faithful “scaled-up” versions in all dimensions, and be made from a material with a constant density so that volume can be determined simply by measuring weight. 

Systems of objects that fill all these needs are hard to find.  Industrial bolts have proven to be an excellent option that satisfies all the requirements, and they are available at your local hardware store or online through merchants such as McMaster-Carr (  Be sure that the bolts are all made of the same material (such as Grade 8 steel bolts) and have threads that run the entire length of the bolt.  The simplest simple set that will allow scaling comparisons is two sizes of bolts in a 1:2 ratio, with 8 of the smaller bolts and one of the larger bolt.

Compare linear dimensions first by laying two of the smaller bolts end-to-end, and noting that the length of the two is equal to the length of the larger bolt.  The same comparison can be made for the heads of the bolts by placing them side-by-side.  The width of the threaded portions of the bolts are a little harder to compare, but putting two small bolts side-by-side in opposite directions allows you to put the threaded parts together and compare them to the corresponding diameter on the larger bolt.  Or simply note that one of the smaller bolts extends across half the diameter of a larger bolt.

Surface areas for the two bolts can be compared by placing four of the smaller bolt heads together, and noting that they cover approximately the same area as the head on a larger bolt.  Because the bolt heads will not “pack” together in the same shape as the larger head, the students will have to do a bit of estimating.  But it should be clear that two of the smaller bolt heads will not cover the area of one of the larger bolt heads, and that four are a very close match.  The area of the head of the bolt that is twice as large is four times as great.

The surface areas of the threaded portions can be compared by cutting a piece of paper or tape that will just wrap around and completely cover the surface on the larger bolt.  That same piece of paper or tape will wrap twice around the diameter of two small bolts placed end-to end, showing that it takes the threaded area of four small bolts to equal the threaded area of one larger bolt.  Or the piece of paper can be divided into four smaller parts (by cutting or with a pencil), each of which will cover the threaded area of a smaller bolt.   Similar methods can be used to compare the areas of all corresponding parts on the two bolts and show its dependence on the square of the linear dimension.

To measure volume, simply weigh the larger bolt, and note that is does not take two small bolts to equal the weight of one bolt that is twice as large.  Eight of the smaller bolts should be very close in weight to that of the larger bolt, showing the dependence of the volume on the cube of the linear dimension.  Since the bolts are made of steel with a constant density, the measured weight of steel is directly proportional to the volume of the bolt.

A more direct way to compare volume is by displacement of water in a graduated cylinder or other tall, transparent container.  It is a simple matter to show that eight of the smaller bolts displace the same amount of water as one larger bolt.  While a more direct measure of volume, this method is less accurate, more time consuming, and much messier than volume comparisons by weight.

All of the above can be extended to a larger range of bolts so that sizes in a 1:2:3:4 ratio can be compared – see the list in the construction section above.  The smallest of the listed bolts (1/4”-28 by ½”) has some small deviations from scale, due largely to the fact that the failure strength of a bolt varies only with the square of its linear dimension (the “shear strength” is directly proportional to the area of the threads that must be sheared off to make the bolt fail, or the area of the shank that must be snapped to break the bolt), so threads can be proportionally smaller on larger bolts and still be much stronger.  Thread counts also do not scale in exact proportions because of standardized thread counts used by industry.   But for all practical purposes, the accuracy of the scaling factors is more than adequate to show the X2 and X3 relationships.

The simplest and closest comparisons are found using eight ½”-13 by 1” bolts with one 1”-8 by 2” bolt.  But the additional ¼”-28 and ¾”-10 bolts add additional data to the comparisons if the time is available for the extra measurements and comparisons.  In addition, the variations from “ideal” scale can be used to introduce the concept of adjusting factors such as thread size and thread count to make the bolts work properly at different scales. 

Those same factors come into play when creating “giant ants” and other creatures of science fiction legend, except that in these cases the scaling ratios are often 1:1000 or higher.  The adjustments that would be necessary for such an overgrown insect (such as proportionally much thicker legs to have enough area to support the huge volume) would make it unrecognizable as an ant.  The much smaller scaling adjustments that are made to bolts of increasing size can provide a good introduction to concepts like these when they (almost inevitably) arise in class.



Parts List

Equipment and costs required to construct apparatus:



Part number


¼”-28 by ½” bolts (27)




½”–13 by 1” bolts (8)




¾”-10 by 1 ½” bolts (3)




1”-8 by 2” bolts (1)




Metal rods

Home Depot



Tape, paper

Grocery store etc.



Total Cost