Constructing Angles with Compasses
≈ 34 minConstructing Angles with Compasses
A protractor is read by eye and is accurate to perhaps half a degree. A constructed angle is exact, because it comes from the geometry of the equilateral triangle and the square.
60° is the foundation. An arc from a point on the line, then the same radius stepped along that arc, gives a chord equal to the radius — an equilateral triangle, so the angle is exactly 60°.
Everything else follows by bisection and combination:
- 30° = bisect 60°
- 15° = bisect 30°
- 90° = perpendicular (or 60° + 30°)
- 45° = bisect 90°
- 120° = two 60° steps along the same arc
- 75° = 60° + 15°
Worked example. To construct 105°: build 90° with a perpendicular, then add 15° by bisecting a 30° twice over. Two exact constructions combine into a third exact angle, with no protractor involved.
Core checkpoint: Build from 60° and 90°, then bisect and add. Any angle that is a combination of these can be constructed exactly.
Why does stepping the compass radius once along an arc of the same radius produce exactly 60°?
You have constructed an exact 60° angle and bisected it twice. What angle in degrees do you now have?
Which combination of exact constructions gives 75°?
A learner draws the arc with one compass setting, then widens the compasses before stepping along it. Why is the resulting angle not 60°?