Equilateral Triangle

Pronunciation: /ˌi kwəˈlæ tər əl ˈtraɪˌæŋ gəl/ ?

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Manipulative 1: Equilateral Triangle. Created with GeoGebra.

An equilateral triangle is a triangle where all of the sides are the same length.[1] Click on the blue points in manipulative 1 and drag them to change the figure.

Properties of an Equilateral Triangle
PropertyEquationDescription
Length of the sideslength of AB = length of BC = length of CA.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.
Anglesangle CAB is congruent with angle ABC and both are congruent with angle BCAThe angles of an equilateral triangle are congruent.
Measure of the anglesmeasure of angle CAB = measure of angle ABC = measure of angle BCA = 60 degrees = pi/3 radians.The angles of an equilateral triangle all measure 60° or π/3 radians.
Altitudeh = s*sin(60°) = s*((square root of 3)/2)
Equilateral triangle showing the calculation of height as h=s*sin(60).The altitude of an equilateral triangle is h=s*sin(60°).
AreaA = (1/2)bh = \(1/2)s*(s*sin(60°)) = (s^2*square root of 3)/4 which is approximately 0.433*s^2The area of a triangle is A=(1/2)bh. Using the formula for the altitude, A=s^2*(square root of 3)/4..
Inradiusr=(square root of 3)/6*sThe inradius of an equilateral triangle is r=(square root of 6)/6*s.
Circumradiusr=(square root of 3)/3*sThe circumradius of an equilateral triangle is R=(square root of 3)/3*s.
Area of the incircleAr=(1/12)pi*s^2The area of the incircle of an equilateral triangle is Ar=(1/12)pi*s^2.
Area of the circumcircleAR=(1/3)pi*s^2The area of the circumcircle of an equilateral triangle is Ar=(1/3)pi*s^2.
Table 1

Constructing an Equilateral Triangle

StepExampleDescriptionJustification
1 Line segment Start with a line segment
2 Line segment with endpoints labeled A and B Label the end points A and B.
3 Construct a circle with center at A and radius AB. 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 center at B and radius AB. 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. Label an intersection of the two circles C.
6 Construct a line segment AC and BC. This forms triangle ABC. 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

  1. equilateral triangle. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-25). http://www.merriam-webster.com/dictionary/equilateral triangle.
  2. 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.. Hodges, Figgis & Co., 1890. (Accessed: 2010-01-02). http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/8/mode/1up/search/equilateral.
  3. Le Clerc, Sébastien; Nattes, John Claude; Pyne, W. H.. Nattes's practical geometry, or, Introduction to perspective, pg 11. W. Miller, 1805. (Accessed: 2010-01-25). http://www.archive.org/stream/nattesspractical00leclrich#page/11/mode/1up/search/equilateral.

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Equilateral Triangle. 2010-01-26. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equilateraltriangle.html.

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Revision History


2010-01-26: Added "References" (McAdams, David.)
2008-12-18: Added 'Properties of an Equilateral Triangle' and changed figure to manipulative (McAdams, David.)
2007-08-20: Added construction of an equilateral triangle (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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