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