Pronunciation: /ɪnˈvɛər.i.ənt/ Explain

Properties of a mathematical object undergoing a transformation that are unchanged by the transformation are called invariant. Manipulative 1 shows a dilation of a triangle. Properties of a triangle that are invariant under dilation are the measures of the angles, and the ratios between the measures of the sides. Properties of the triangle that are not invariant are the lengths of the sides and the area of the triangle.

Click on points and drag them to change the figure.

Which of the following never change for dilation: Angle measure, length of side, ratio of sides, are of the triangle?
Manipulative 1 - Invariant Created with GeoGebra.


  1. McAdams, David E.. All Math Words Dictionary, invariant. 2nd Classroom edition 20150108-4799968. pg 101. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Glenn, Oliver E.. A Treatise on the Theory of Invariants. Ginn and Company. 1915. Last Accessed 8/7/2018. Buy the book
  3. Lin I-hsiung. Geometric Linear Algebra. vol 2 pg 194-205. World Scientific Publishing Company. May 6, 2008. Last Accessed 8/7/2018. Buy the book
  4. A. B. Kharazishvili. Transformation Groups and Invariant Measures: Set-Theoretical Aspects. World Scientific Publishing Company. September 1998. Last Accessed 8/7/2018. Buy the book

Cite this article as:

McAdams, David E. Invariant. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
3/2/2010: Added "References", clarified wording. (McAdams, David E.)
9/16/2008: Initial Version. (McAdams, David E.)

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