Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles

(Proposition 21 from Book 11 of Euclid's “Elements”)

Any solid angle is contained by rectilinear angles (whose sum is) less [than] four right angles.

Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1
Propositions: 2

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016


  1. This proposition is only proved for the case of a solid angle contained by three rectilinear angles. However, the generalization to a solid angle contained by more than three rectilinear angles is straightforward (translator's note).