# 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.

Properties of an Equilateral Triangle
PropertyEquationDescription
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
1. McAdams, David E.. All Math Words Dictionary, equilateral triangle. 2nd Classroom edition 20150108-4799968. pg 71. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. equilateral triangle. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/9/2018. http://www.merriam-webster.com/dictionary/equilateral triangle. Buy the book
3. Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid. pp 8,11-12. 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
4. 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

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.

### 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.)