Trisection — Definition, Formula & Examples

Trisection is the act of dividing an angle or a line segment into three equal parts. While trisecting a line segment with a compass and straightedge is straightforward, trisecting an arbitrary angle using only those tools was proven impossible in 1837.

To trisect a geometric object is to partition it into three congruent sub-parts. For a line segment $\overline{AB}$ , trisection produces points $P$ and $Q$ such that $AP = PQ = QB$ . For an angle $\angle AOB$ of measure $\theta$ , trisection yields two rays that divide it into three angles each measuring $\tfrac{\theta}{3}$ .

How It Works

Trisecting a line segment is done by construction: you can use parallel lines or repeated compass arcs to mark three equal lengths along an auxiliary line, then project those divisions back onto the original segment. Trisecting an angle is a different story. Pierre Wantzel proved in 1837 that no general compass-and-straightedge method can trisect every angle. Specific angles like

90°

can be trisected (since

30°

is constructible), but a general

60°

angle cannot be trisected this way because

20°

is not a constructible angle. Other tools — such as a marked ruler (neusis) or origami folds — can accomplish angle trisection.

Worked Example

Problem: Trisect the line segment from

A(0, 0)

B(9, 0)

. Find the two trisection points.

Step 1: Divide the total length into three equal parts.

AB = 9, \quad \frac{AB}{3} = 3

Step 2: Starting from

A

, place the first point at distance 3 and the second at distance 6.

P = (3, 0), \quad Q = (6, 0)

Step 3: Verify the three segments are equal.

AP = PQ = QB = 3

Answer: The trisection points are

P(3, 0)

and

Q(6, 0)

Why It Matters

The impossibility of angle trisection is one of the three famous problems of ancient Greek geometry and appears in many high school and college geometry courses. Understanding why it fails deepens your grasp of what compass-and-straightedge constructions can and cannot achieve, which connects to abstract algebra and the theory of constructible numbers.

Common Mistakes

Mistake: Assuming that because you can bisect any angle, you can trisect any angle with compass and straightedge.

Correction: Bisection and trisection are fundamentally different operations. Wantzel's proof shows that trisecting a general angle (like 60°) is impossible with compass and straightedge alone. Only certain special angles can be trisected.