Equilateral Triangle
Pronunciation: /ˌi.kwəˈlæ.tər.əl ˈtraɪˌæŋ.gəl/ Explain
The sides of this triangle are always the same length, are equilateral.
The sides of this triangle are always the same length, are equilateral.
 Manipulative 1  Equilateral Triangle Created with GeoGebra. 

An equilateral triangle is a triangle where all of the
sides are the same length.^{[3]}

Properties of an Equilateral Triangle
Property  Equation  Description 
Length of the sides   The length of the sides of an equilateral triangle are equal by definition. The length of a side of an equilateral triangle is conventionally represented by the variable s. 
Angles   The angles of an equilateral triangle are congruent. 
Measure of the angles   The angles of an equilateral triangle all measure 60° or . 
Altitude    The altitude of an equilateral triangle is h = s·sin(60°). 

Area   The area of an equilateral triangle is . Using the formula for the altitude, . 
Inradius   The inradius of an equilateral triangle is . 
Circumradius   The circumradius of an equilateral triangle is . 
Area of the incircle   The area of the incircle of an equilateral triangle is . 
Area of the circumcircle   The area of the circumcircle of an equilateral triangle is . 
Table 1 
Constructing an Equilateral Triangle
Step  Example  Description  Justification 
1 

Start with a line segment 

2 

Label the end points A and B. 

3 

Construct a circle with the center at A and the radius the length of segment AB. 
Euclid's Elements Book 1 Postulate 3: A circle can be drawn with any center and any radius. 
4 

Construct a circle with the center at B and the radius the length of segment AB. 
Euclid's Elements Book 1 Postulate 3: A circle can be drawn with any center and any radius. 
5 

Label an intersection of the two circles C. 

6 

Construct a line segment AC and BC. This forms triangle ABC. 
Euclid's Elements Book 1 Postulate 1: A straight line can be drawn from any point to any point. 
7 
Since they are radii of the same circle, the line segment AC
is the same length as the line segment AB. Similarly, the
line segment BC is the same length as the line segment
AB. 
Euclid's Elements Book 1 Definition 15: A circle is all points equidistant from a center point.

8 
Since AB ≡ AC
and AB ≡ BC, then it must be
true that AC ≡ BC. 
Euclid's Elements Book 1 Common Notion 1: If A = B and B = C then A = C.

9 
By the definition of an equilateral triangle, the triangle
ABC is equilateral. 
Euclid's Elements Book 1 Definition 20: An equilateral triangle is a triangle where the length of all three sides is the same.



Q.E.D. 

Table 2 
References
 McAdams, David E.. All Math Words Dictionary, equilateral triangle. 2nd Classroom edition 201501084799968. pg 71. Life is a Story Problem LLC. January 8, 2015. Buy the book
 equilateral triangle. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 7/9/2018. http://www.merriamwebster.com/dictionary/equilateral triangle. Buy the book
 Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid. pp 8,1112. Translated by Casey, John, LL.D. F.R.S.. www.archive.org. Hodges, Figgis & Co.. 1890. Last Accessed 7/9/2018. http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/8/mode/1up/search/equilateral. Buy the book
 Le Clerc, Sébastien; Nattes, John Claude; Pyne, W. H.. Nattes's practical geometry, or, Introduction to perspective. pg 11. www.archive.org. W. Miller. 1805. Last Accessed 7/9/2018. http://www.archive.org/stream/nattesspractical00leclrich#page/11/mode/1up/search/equilateral. Buy the book
More Information
Cite this article as:
McAdams, David E. Equilateral Triangle. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equilateraltriangle.html.
Image Credits
Revision History
4/20/2019: Updated expressions and equations to match new format. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
1/26/2010: Added "References". (
McAdams, David E.)
12/18/2008: Added 'Properties of an Equilateral Triangle' and changed figure to manipulative. (
McAdams, David E.)
8/20/2007: Added construction of an equilateral triangle. (
McAdams, David E.)
7/12/2007: Initial version. (
McAdams, David E.)