666 Six Pointed Star

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Star of the Kenites

Equilateral Triangle

DOWNLOAD Mathematica Notebook EXPLORE THIS TOPIC IN the MathWorld ClassroomEquilateralTriangle

An equilateral triangle is a triangle with all three sides of equal length a, corresponding to what could also be known as a “regular” triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal 60 degrees angles.

The altitude h of an equilateral triangle is

 h=asin60 degrees=1/2sqrt(3)a,
(1)

where a is the side length, so the area is

 A=1/2ah=1/4sqrt(3)a^2.
(2)

EqTriangleInscribeCirc

The inradius rcircumradius R, and area A can be computed directly from the formulas for a general regular polygon with side length a and n=3 sides,

r = 1/2acot(pi/3)
(3)
= 1/2atan(pi/6)
(4)
= 1/6sqrt(3)a
(5)
R = 1/2acsc(pi/3)
(6)
= 1/2asec(pi/6)
(7)
= 1/3sqrt(3)a
(8)
A = 1/4na^2cot(pi/3)
(9)
= 1/4sqrt(3)a^2.
(10)

The areas of the incircle and circumcircle are

A_r = pir^2
(11)
= 1/(12)pia^2
(12)
A_R = piR^2
(13)
= 1/3pia^2.
(14)

Central triangles that are equilateral include the circumnormal trianglecircumtangential trianglefirst Morley triangleinner Napoleon triangleouter Napoleon trianglesecond Morley triangleStammler triangle, and third Morley triangle.

EquilateralTriangleEquation

An equation giving an equilateral triangle with R=1 is given by

 max(-2y,y-xsqrt(3),y+xsqrt(3))=1.
(15)

EquilateralTriangleConst

Geometric construction of an equilateral consists of drawing a diameter of a circle OP_O and then constructing its perpendicular bisector P_3OB. Bisect OB in point D, and extend the line P_1P_2 through D. The resulting figure P_1P_2P_3 is then an equilateral triangle. An equilateral triangle may also be constructed from the intersections of the angle trisectors of the three interior angles of any triangles (Morley’s theorem).

Napoleon’s theorem states that if three equilateral triangles are drawn on the legs of any triangle (either all drawn inwards or outwards) and the centers of these triangles are connected, the result is another equilateral triangle.

Given the distances of a point from the three corners of an equilateral triangle, ab, and c, the length of a side s is given by

 3(a^4+b^4+c^4+s^4)=(a^2+b^2+c^2+s^2)^2
(16)

(Gardner 1977, pp. 56-57 and 63). There are infinitely many solutions for which ab, and c are integers. In these cases, one of abc, and s is divisible by 3, one by 5, one by 7, and one by 8 (Guy 1994, p. 183).

Begin with an arbitrary triangle and find the excentral triangle. Then find the excentral triangle of that triangle, and so on. Then the resulting triangle approaches an equilateral triangle. The only rational triangle is the equilateral triangle (Conway and Guy 1996). A polyhedron composed of only equilateral triangles is known as a deltahedron.

TriangleCircumRect

Let any rectangle be circumscribed about an equilateral triangle. Then

 X+Y=Z,
(17)

where XY, and Z are the areas of the triangles in the figure (Honsberger 1985).

EqTriangleSquare

The smallest equilateral triangle which can be inscribed in a unit square (left figure) has side length and area

s = 1
(18)
A = 1/4sqrt(3) approx 0.4330.
(19)

The largest equilateral triangle which can be inscribed (right figure) is oriented at an angle of 15 degrees and has side length and area

s = sec(15 degrees)=sqrt(6)-sqrt(2)
(20)
A = 2sqrt(3)-3 approx 0.4641
(21)

(Madachy 1979).

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