TOPICS

# Equilateral Triangle

An equilateral triangle is a triangle with all three sides of equal length , 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 angles.

The altitude of an equilateral triangle is

 (1)

where is the side length, so the area is

 (2)

The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon with side length and sides,

 (3) (4) (5) (6) (7) (8) (9) (10)

The areas of the incircle and circumcircle are

 (11) (12) (13) (14)

Central triangles that are equilateral include the circumnormal triangle, circumtangential triangle, first Morley triangle, inner Napoleon triangle, outer Napoleon triangle, second Morley triangle, Stammler triangle, and third Morley triangle.

An equation giving an equilateral triangle with is given by

 (15)

Geometric construction of an equilateral consists of drawing a diameter of a circle and then constructing its perpendicular bisector . Bisect in point , and extend the line through . The resulting figure 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, , , and , the length of a side is given by

 (16)

(Gardner 1977, pp. 56-57 and 63). There are infinitely many solutions for which , , and are integers. In these cases, one of , , , and 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.

Let any rectangle be circumscribed about an equilateral triangle. Then

 (17)

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

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

 (18) (19)

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

 (20) (21)

Triangle line picking for points in an equilateral triangle with side lengths gives a mean line segment length of

 (22) (23)

30-60-90 Triangle, Acute Triangle, Deltahedron, Equilic Quadrilateral, Fermat Points, Gyroelongated Square Dipyramid, Icosahedron, Isosceles Right Triangle, Isosceles Triangle, Morley's Theorem, Octahedron, Pentagonal Dipyramid, Polygon, Regular Polygon, Reuleaux Triangle, Right Triangle, Scalene Triangle, Snub Disphenoid, Tetrahedron, Triangle, Triangle Packing, Triangular Dipyramid, Triaugmented Triangular Prism, Viviani's Theorem Explore this topic in the MathWorld classroom

## Explore with Wolfram|Alpha

More things to try:

## References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.Conway, J. H. and Guy, R. K. "The Only Rational Triangle." In The Book of Numbers. New York: Springer-Verlag, pp. 201 and 228-239, 1996.Dixon, R. Mathographics. New York: Dover, p. 33, 1991.Fukagawa, H. and Pedoe, D. "Circles and Equilateral Triangles." §2.1 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 23-25 and 100-102, 1989.Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, 1977.Guy, R. K. "Rational Distances from the Corners of a Square." §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181-185, 1994.Honsberger, R. "Equilateral Triangles." Ch. 3 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., 1973.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 19-21, 1985.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 115 and 129-131, 1979.

## Cite this as:

Weisstein, Eric W. "Equilateral Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquilateralTriangle.html