Painting - Square Root of Pi

This oil painting is an original construction of Crockett Johnson, and proceeds from the assumption that the circle has been "squared." If the circle has radius one, and if square with the same center has the same area, Crockett Johnson argued that the inscribed rectangle shown, which has a diagonal that meets opposite points of intersection of the square and circle, has an area equal to the square root of pi.
The verification of Crockett Johnson's construction is straightforward. The circle has radius one so that its area is pi. Because it is assumed that the circle has been "squared," the area of the square is also pi, and the length of one of its sides equals the square root of pi. The area of the rectangle is equal to the sum of the area of the two triangles formed by the diagonal. These triangles have bases equal to the diameter of the circle (2) and height equal to half the length of the side of the square (half of the square root of two). Hence each triangle has area half of the square root of pi, and the entire rectangle has area equal to the square root of pi. There is a second rectangle in the painting of the same area.
There are two paintings in the collection with this title. The geometry of the two is identical; only the dimensions and colors are different. For this painting, #100 in the series, Johnson illustrates the subject vividly through the electric blue color of the rectangle with area equal to the square root of pi. Its partner, #89 in the series (1979.1093.58), displays the same rectangle in white, which contrasts brilliantly with its black and purple surroundings.
This painting is unsigned and its precise date is unknown. It has a plain wooden frame.
Currently not on view
Object Name
date made
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 83.8 cm x 83.8 cm x 4.2 cm; 33 in x 33 in x 1 5/8 in
ID Number
catalog number
accession number
Science & Mathematics
Crockett Johnson
See more items in
Medicine and Science: Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center
Credit Line
Ruth Krauss in memory of Crockett Johnson
Additional Media

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