Ruler and Compass Impossibility

Key idea

Trisecting angles and doubling the cube are impossible.

In depth

Constructible lengths lie in towers of quadratic extensions, so they have degree a power of 2 over Q. Hence doubling the cube (∛2), trisecting a general angle, and squaring the circle (π transcendental) are impossible.

Worked example

∛2 has degree 3 over Q, which is not a power of 2, so cannot be constructed.

Real-world uses

• Explaining why some geometric constructions are impossible with only a ruler and compass

Common mistake

Believing the problems are unsolved rather than provably impossible with the given tools.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Differential Equations, Abstract Algebra & Applied Math

• Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
• Runge–Kutta Methods – Higher-accuracy numerical solvers.
• Second-Order Homogeneous Equations – Use the characteristic equation.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.