The Wonders And Formulas Of Equilateral Triangles In Mathematics And Geometry

Equilateral Triangle

A triangle with all equal sides and all equal angles

An equilateral triangle is a three-sided polygon in which all three sides are equal in length and all three angles are equal to 60 degrees. This means that the triangle has rotational symmetry of order 3 or in other words, it can be rotated by 120 degrees and still look the same.

Properties of an equilateral triangle:

1. All three sides are of equal length.
2. All three angles are equal to 60 degrees.
3. The sum of the angles is 180 degrees.
4. It has a rotational symmetry of order 3.
5. The angle bisectors of an equilateral triangle intersect at the centre of the circumscribed circle.
6. The circumcenter and incenter coincide in an equilateral triangle.
7. The ratio of the length of the altitude of an equilateral triangle to the length of a side is equal to sqrt(3)/2.

Some important formulas related to the equilateral triangle are:

1. The perimeter of an equilateral triangle is 3 times the length of one side, P = 3s where s is the length of each side.
2. The area of an equilateral triangle is (sqrt(3)/4) times the square of the side length, A = (sqrt(3)/4) * s^2.
3. The height of an equilateral triangle is (sqrt(3)/2) times the length of a side, h = (sqrt(3)/2) * s.

The equilateral triangle has many interesting properties used in mathematics, engineering and geometry. It is used to construct regular polygons, as the basis of the sierpinski triangle, and in the study of fractals and tessellations.

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