Welcome to Aquarius, Volume 8 (August 16, 2007), Squaring the Circle Exactly.

Summary of Constructions that accomplish squaring the circle exactly. (2 pages)

Introduction to constructions (for beginners):

Re-construction of angle

Re-construction of similar triangle

Square root of a value, squaring the rectangle

The 26 constructions performed to accomplish squaring the circle exactly:

The five-sixths constructions:

Construction #1 (tan=2)

--->(to get 1, 1/5, 1/6 and 5/6 values, together with Construction #2)

Construction #2 (tan=2)

--->(to get 1, 1/5, 1/6 and 5/6 values, together with Construction #1)

Construction #3 (tan = 5/6) --->[to get (5/6) and (6/5) and other helpful values]

The nine-elevenths constructions:

Construction #4 (tan= 9/11) --->[ to get (9/11) and (11/9)]

The point fifty-five constructions:

Construction #5 (tan= 0.55) --->(to get 0.55 value)

Construction #6 (tan= 0.45) --->(to get 0.45 value)

The 36-54 rt tri, the 18-72 rt tri:

Construction #7 --->(Construction of the 72-degree angle)

Construction #8 --->(Construction of the 18-72 degree right triangle)

Dividing line, new constructions specifically required to square the circle follow:

The (secant 18)^4 constructions:

Construction #9 --->[to get line length 1.222291236..., (secant 18)^4]

Construction #10 --->(to get 7.778, and 2.222, 0.7778 and 0.2222)

Study of the next section, The Definition of MQ, is strongly recommended in order for the reader to understand the purpose of the constructions that follow.

Construction of TN and NT:

Construction #11-1, 11-2 --->(Construction of TN and NT in two steps)

Construction of MQ and M: [Constructions #12, 13, 14, 15, 16]

Construction #12 --->(six steps, to get NTP value [0.999937891...])

Construction #13 --->(to get tangent = 1.222291236...)

Construction #14 --->( to get 1.22221532123408073944164930824006... [MQ+0.2222])

Construction #15 --->(final construction to get MQ), simple line constructions on page

Construction #16 (square the rectangle, by reference to this procedure)

The Get Pi constructions:

Construction #17-1 --->(to get ratio of 1.618033989...)

Construction #17-2 --->(to get line length of Phi^2)

Construction #18 --->(to get tangent = [Pi*(5/6)] )

Construction #19 --->(to get line length [Pi*(5/6)] )

Construction #20 --->(in two steps, to get line length of Pi)

The Pi and 1 constructions (copy from Volume 1):

(21) PM constructions [(Pi/2-1) is cosine]

(22) RP construction [(Square Root of Pi)/2 is cosine]

(23) OR construction [1/sqrt(Pi) is cosine]

(24) RE construction [1/sqrt(Pi) is length of Base]

(25) Square to Circle (STC)

(26) Circle to Square (CTS)

The Pi and MQ constructions --->(alternative square = circle area, text and equations)

Link to: (Welcome) or (Geometry Alpha Index).