Drawing the Square and its Circle - Ad Infinitum

 

Note: updated for clarity 9/26/2026 

Serlio's cube:  It's on the frontispiece of Sebastiano Serlio's On Architecture *, published c.1550, translated into English in 1611. The geometry shows the structure for adding a circle within a square - for example, a dome above a square space. 

This lower section of the book's cover shows the tools a master builder would have used for design c. 1550.  The cube is a layout tool.

 Serlio's geometric diagram: a base of a Tuscan column. **

 

And his diagram for a Corinthian column. **

This is his drawing of a Roman temple; it is design and structure is based on the circle and its square. 

He included the plan.  For more details see my blog post on the temple **** 

 

How can these layouts be drawn with a compass and a straight edge?

 

The first step is the geometry of the square.

The diagram is in Audels' Carpenters and Builders Guide.  published in 1928***. 

Next: add the circles that are larger and/or smaller than the square. Below is a sequence.

1) the square with its diagonals

2) Half the length of the square's diagonal is the radius for a circle around the square.  

3) The mid-point of a side of the square gives the length of the  radius for the circle within the square. 

4) The points where the inner circle crosses the square's diagonals determine a square within the first square.                               

 Reader beware: This could go on ad-infinitum. 

 

 

What if the beginning dimension is the diameter of a column?  How can a square with the right proportions - the pedestal, for example - be drawn around that circle?  

The diagram is from Edward Shaw's Civil Architecture* Plate 2. 'Problem X: drawing a tangent to a given circle'.

 

The length of the sides of the square is the diameter of the circle. But where does that length 'sit' on the circle? 

The radius of the circle (C-A) is half its diameter (D-E). By definition it makes a point at the circumference of the circle (A).     

1) C is the center of the circle. C-A is its radius.

2)  C'-A is also the length of the radius. A tangent to the circle, C'-A-C' is the length of the diameter.  It can be 'trued' using step 3. 

3) The diameter has a length, therefore a point at each end   Both C's can be one end of a line which extends to the point C, (the center of the circle). If they are equal, the line C'-A-C' is true, sitting symmetrically on the circle's circumference.  

4) Those lines, extended, create 2 more C'-A-C's. The length of those 2 lines is the diameter of the circle: 2 more points. The points can be joined to become the square around the circle.    

5)  C- C' becomes the radius for a circle around the square, around the circle.

Too many words! It's much easier just to draw it together.

I've written other blog posts about the use of this geometry in real construction.  

Practical Geometry at MIT, April 9, 2018.

**** Serlio studies a Roman Temple, April 1, 2023

The Perfect Cube and its Sphere, July 16, 2024 

James Gibbs' Steeples, February 25, 2022    This is squares, not circles.

* Audels Carpenters and Builders Guide, # 2, NYC, 1923, p.494

** Sebastiano Serlio, On Architecture, c. 1550; translated from Italian into English by Hart and HIcks,1996.  illustrations on pages 125, 257, 341 

*** Edward Shaw, Civil Architecture, 6th edition, published in 1852

 

Owen Biddle's 'Young Carpenter's Assistant' , Plate I, G

A note on Owen Biddle's Plate I, Diagram G. in his pattern book for beginning carpenters. *

 

I wrote about Diagram G on this post: https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

I said that Biddle was not just introducing his 'carpenter assistant' to geometry; in Diagram G Biddle was explaining how to layout a square corner to work out a structural detail, cut a board, or set a frame on site.

Since then I have explored the theoretical geometry of that diagram.

The number of right angles which can be drawn in a circle is infinite. The rule always works. That understanding is part of why geometry is seen as mystical or sacred.

This 'squaring the circle' diagram is from

Robert Lawlor's Sacred Geometry*. (page 77, diagram 7.5)

It uses a geometry similar geometry to Biddle's diagram G: a diameter and an angle. Here the diameters are evenly spaced and the same angle  is used at every point on the circumference. But the angle is not 90*. It is not a 'square angle'.

This is decorative, not structural.

The shapes do not close. The line continues for 5 rotations. It does not create a square, but seeks to define the perimeter of a circle with straight lines. 

,

I am often told that I work with Sacred Geometry, that the geometric patterns I recover are theoretical, mystical, and sacred. I agree they are geometry. No, they are not sacred. They are practical. They are geometry used in construction.

Here is how Biddle's diagram comes about: 

Begin with  a point  - A

Choose a radius - A-B,  and draw a circle. Using the daisy wheel find the diameter - B- A- C, dotted and dashed line.

Pick a point on the circumference of the circle - D.

Here I have chosen 3 different D's  at random.

Connect B-D and D-C.

Each diagram will have a 90* (right) angle at the intersection of  B-D-C.

Wherever the D is placed. the angle will be 90*.

Biddle's Diagram G begins with my line B-D.

It describes how to find my 90* angle of B-D-C. (his a-b-c) The answer is to find the diameter of a circle (a-d-c) that intersects a. That will give c. That will give the 90* the carpenter needs.

 

By Hound and Eye* has a very similar diagram for drawing a right angle .
The book is a  guide to furniture design, full of practical geometry. Each geometric problem is described step by step; practice work sheets are included.

This pattern is the beginning of a handmade try square. 

 

*Owen Biddle's The Young Carpenter's Assistant, 1805, Philadelphia. Dover Publishing  reprint,  See my Bibliography for more information.

*Robert Lawlor, Sacred Geometry, Philosophy and Practice, 1982, Thames and Hudson, London.

*Geo.R. Walker & Jim Tolpin, By Hound and Eye, A Plain & Easy Guide to Designing Furniture with No Further Trouble, 2013,  Lost Art Press, Kentucky The diagram shown above is from page 57.

This pattern is 4 overlapping hexagons.

My granddaughter, who is 7, watched me add the images to this post.

She wanted us to 'square the circle'. I did, using right angles where the diameters met the circumference. That produced these overlapping 6 hexagons, not squares.

She watched closely and observed that accurate work was not easy: my lines did not always cross exactly in the center of the circle. When we finished she asked me to erase all the diameters. This is the result. Maybe she will show me later what she added to the copy I printed for her.