By Jim Fonseca
There are several well-known triangles, colloquially called golden triangles, which incorporate some aspect of the mathematical ratio know as phi, 1.61803…, symbolized by the Greek letter Φ. The triangles are called ‘golden’ because that mathematical constant is referred to as the Golden Ratio. That ratio comes about from the idea of cutting a line into two pieces such that the ratio of the two pieces to each other equals the ratio of the larger piece to the original whole piece.
An example helps. So think of the Golden Ratio this way. You have a wooden yardstick, 36 inches. Measure off 22¼ inches and cut it. The remaining short piece will be 13¾ inches. Now do the math: 36/22.25 is 1.618. But 22.25/13.75 is also 1.618. That’s the Golden Ratio. The Golden Ratio is the only way you can cut the stick to satisfy those initial conditions. That’s what makes it so special. And like π, the value of Φ, 1.61803…., is an endless non-repeating fraction.
Previous studies of Golden Triangles have always started with the sides of triangles, probably because we learn about Φ in terms of length and pieces, like in the example I just gave. But in a recent article on ResearchGate, I proposed a new approach to looking for relationships in triangles by starting with the angles of a triangle. After all, while mathematicians may talk about trilaterals, in everyday conversation everyone else speaks of triangles. You can see the full research article including references here
We know the sum of the angles of a triangle equals 180°. Once we know of the Φ relationship and its importance in many aspects of geometry, it seems perfectly logical to ask the simple question: “If the three angles of a triangle were in phi proportion to each other what would these angles be?”
Based on how we calculate phi the formula to answer this question becomes
This simplifies to
When we solve the formula, x is precisely 90°. The other two angles as shown in Figure 1 are approximately 55.623° (angle A) and 34.377° (angle C). If we do the math for the angles, B/A and A/C each gives us…