Yes.
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A regular polygon is any polygon that has sides which are the same length and angles whose measures are equal. An equilateral triangle (also equiangular triangle) is a regular polygon. Other isosceles triangles (equilateral triangles are isosceles, but they are an exception) and scalene triangles are not regular polygons. A side note: Only in a triangle is a polygon regular solely if it is equilateral. (Since an equilateral triangle is equiangular as well). This is NOT always true in other polygons, like quadrilaterals, where it can be equilateral but not necessarily equiangular (a rhombus) or equiangular but not equilateral (a rectangle).
An equilateral triangle is a regular shape because it has 3 equal sides.
If by regular, you're referring to a regular polygon which has all sides equal length, and all angles equal, then an equilateral triangle is a regular triangle. If you draw a scalene triangle or an isosceles triangle then it will not be equilateral.
In our example, the area of the equilateral triangle is 1/6 of the area of the regular hexagon
equilateral triangle is an example of regular polygon