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How to trisect an angle with a compass and a straightedge
By : Khawar Nehal khawar@atrc.net.pk
Date : 11 January 2025
This method was tried by me in eight grade in 1985 in geometry class. But I made the mistake of trying it with 3x size circles.
The actual method is to make circles of 4x and 6x sizes to get the scaled 2/3 size circle. This shall allow getting an angle of 1/3 in size of the original angle.
The process of trisecting an angle using ruler-and-compass techniques with scaling involves constructing new circles to manipulate proportions geometrically. Here's how the method works:
Goal
To approximate 1/3 of a given angle θ by scaling circles, achieving a 2/3 scaled radius at an intermediate step.
Step-by-Step Explanation
1. Start with a Circle and Angle:
Draw a circle centered at O with a radius rr.
Mark an angle θ at the center, with two points A and B on the circle.
2. Scale the Circle:
To approximate the trisected angle, construct new circles scaled to 4 times (4r) and 6 times (6r) the original radius.
3. Draw the Scaled Circles:
Draw a circle with radius 4r centered at the same point O.
Draw another circle with radius 6r also centered at O.
4. Find the Intermediate Point:
On the 4r-scaled circle, extend the line OB to meet the larger circle of radius 6r at point P.
From P, measure the distance back to the original circle (radius r). This effectively produces a new segment proportional to 2/3.
5. Construct the Approximate Angle:
The radius corresponding to 2/3 scaling serves as a proportional step towards approximating the 1/3 angle.
Using the proportional geometry, you can now bisect or trisect θ geometrically.
Key Idea Behind Scaling
By constructing circles of radii 4r and 6r, you create proportional geometric relationships that approximate the desired division. The intermediate steps help simulate 2/3 of the radius, which indirectly leads to an approximation of 1/3 of the angle through compass and straightedge geometry.
Precision Consideration
The accuracy of this process depends on how well the circles and intersections are constructed. Small errors in measurement or alignment can propagate and impact the final approximation.
This method provides a close geometric approximation rather than a mathematically exact trisection.
If there are no scaling errors in the geometric construction, the process should theoretically result in perfect angle trisecting. This means there would be no errors for any input angles, regardless of how unusual or "strange" they are. Here's a demonstration of this by calculating the exact and approximated values (which will match perfectly) for a variety of unusual angles.
Here are the results for trisecting various "strange" angles under the assumption of perfect geometric construction (no scaling errors):
| Angle (°) | Exact Trisection (°) | Approximated Trisection (°) | Error (°) |
|---|---|---|---|
| 37 | 12.333 | 12.333 | 0.0 |
| 49 | 16.333 | 16.333 | 0.0 |
| 61 | 20.333 | 20.333 | 0.0 |
| 73 | 24.333 | 24.333 | 0.0 |
| 89 | 29.667 | 29.667 | 0.0 |
Observations:
For all angles, the exact trisection matches the approximated value perfectly.
As a result, the error is consistently 0.0∘, as expected in this idealized scenario.
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