Determine the Mean Proportion or Square Root Geometrically

In case you ever find yourself in the woods with a compass, straightedge, pencil and paper, and a 10 cm ruler, here is how to find the square root or mean proportion.


The Tutorial

  1. Understand the terms. Usually, what is meant by a "mean proportional", is a line (or number) between two lines in length such that its proportion over the first shorter line is the same proportion the second longer line has to it -- thus it is the mean, And what is meant by "square root" is that number (or line) which when multiplied times itself, produces a square; however, the root itself may be the product or sum of two or more other numbers. The method presented here takes advantage of exactly that fact, so if you have a square in mind, please make it the product of a number a and a number b.
  2. Scale up your numbers. If you are finding the square root of a large number, you will need to scale up your centimeter or millimeter ruler, e.g. if your number = 6085 and you want to find the square root of it, then your 100 divisions of a 10 centimeter ruler instead of meaning 100, would seem to need to mean 10,000, so each division is worth 100 more divisions, since 100*100= 10,000. You would find the number 6096 just shy of 6.1 centimeters then, right? But what you really want to measure in are the factors of 6096, and those will be less than 100 x 100, so let's say each division on your 100 millimeter ruler is worth 10, which gives us the number of 1,000 as our new yardstick.
  3. Choose factors a and b of 6096 as a=127, b=48 and center = (127+48)/2= 87.5; that's perfectly acceptable. Mark point a at the far left, point b 127 units to a's right, and point c at the other endpoint, 48 more units to the right, with vertical hash marks. Also mark the center of the line with a hash mark at 87.5 units.
  4. Describe the semicircle above labeled points a,b and c of line ac containing the segments ab and bc equaling 127 and 48 respectively.
  5. Draw line "BD" at right angles to line ac up to intersect the circle at point d. Join lines ad and dc.
  6. At point d, write "d/M" and at point b. change it to read "b/P" where "MP" means "Mean Proportional".
  7. Now since angle "ADC" is an angle in a semicircle, it is right. And since in the right angle "ADC", "DB" (or MP) has been drawn from the right angle perpendicular to the base, therefore "DB" is a mean proportional between the the segments of the base, "AB" and "BC". If you now measure "DB" (or MP), you should find it equals 78.0769, which is not only the square root of 6096 as you originally had sought, and the root of its its factors 127 and 48, but 78.0769 / 48 = 1.6266 and 127 / 78.0769 = 1.6266 as well. That is to say, the square root is also the mean proportional!

Explanatory Charts, Diagrams, Photos

  1. Final Image

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the article How to Determine the Geometric Version of the Golden Mean (Ratio or Proportion) for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.


  • If you love these relationships, you are probably already an avid fan of Euclid's "Elements". The proofs are typically very short, extremely logical, and usually only rely on a few former proofs.
  • Extra Credit Assignment: One of the quadratic roots of Phi = {x1, x2} is 1.61803398874989 and the other root is -.61803398874989, the sum of which is 1. The total distance between them however is 1.61803398874989 - -.61803398874989 = 2.23606797749978. See if you can determine the Mean Proportion of 1.00000 and .61803398874989 and the Square Root of their Product, .61803398874989 by using 1+.62 = 1.62 using the above geometric method. For your check figure, the accurate square root of .61803398874989 = 0.78615137775742; 0.78615137775742 / 1 = 0.78615137775742 and 0.78615137775742 / .61803398874989 also = 0.78615137775742, so it is the Mean Proportion and Square Root of the Product of the two numbers, 1 and .61803398874989. See how close you can come for this transcendental number, Phi. By the way, the product of the two roots, 1.61803398874989 and -.61803398874989 = -1, which has the square root of -1 as its its square root, but sqrt(-1) = i, and then we are in the realm of imaginary numbers. And, btw, (1-.61803398874989)*.61803398874989=0.236067977499791 while (1.61803398874989)+(1-.61803398874989) = 2, which together makes that distance referred to above, 2.236067977499791.


  • Perhaps as close as you could get with a ruler was 78. Still, 78^2 = 6084, which is a good approximation in the realm of square roots to the square root of 6096. And why are accurate (or good approximations) square roots important? Because squares FIT so nicely together, and Mother Nature not only likes a good FIT, she likes the niches between approximate FITs to try new possibilities within. FIT is very important in Nature, but so is roughness and approximation as it helps to create space for defensive structures to protect the plant or other life form. Also, grooves can be important, as in the cactus plant, where water can form and run down to the roots -- the grooves being anomalies in the otherwise round arm or trunk of the cactus tree. By being shadowed and more damp, with the wind's effect, they also help cool the plant. Cool and groovy ... the cactus tree -- just be warned of those nasty thorns!
  • Since is 3.46410161513775 / 12 = 0.288675134594812, I gather that an inner side-sector of 28.88675% contains the Area, 3.46410161513775. If Area^2 = Area, then that is the answer to the question. Not such a thorny problem after all, since all that's been said is that the (square root of the Area)^2 = the Area.
  • By goal seeking, it's found that the degree measurement that creates a sector of = 3.46410161513775 sq. in. = 24.8098002939806º, or 28.8486049930007% of 86º. That's the real answer ... until one finds out that 86 was rounded from 85.94366926962350 -- so goal seeking again produces an answer of 24.8098002939806, but 24.8098002939806 / 85.94366926962350 = 28.8675134594812% -- the exact ratio of sqrt(12) to 12 we got before. Now it's definite.
  • We know that a side is the square root of a square, but what is the square root of a cylindrical section or pie sector? Hint: It's the square root of the Area of the Sector, which is given in the diagram.

    So in the Diagram, the Area in square units = 12, therefore sqrt(12) is the square root of the shown sector. Sqrt(12) = 3.46410161513775, or about 86.60% of the radius, 4. That distance times itself equals the Sector's Area.

Related Articles