Classifying two-dimensional shapes

Different categories of triangles - part 2

Second, looking at the sides, we can categorize triangles as being equilateral, isosceles or scalene.

An equilateral (equal-sided) triangle is one in which all three sides are equal. Because of the rigid nature of triangles, the only possibility for an equilateral triangle is one in which the three angles are also equal (to 60 degrees). So an equilateral triangle must be a regular triangle. This is only true of triangles. For example, you can have an equilateral octagon (with eight equal sides) in which the angles are not equal: just imagine joining eight equal strips of card with paper fasteners and manipulating the structure into many different shapes, all of which are equilateral octagons but only one of which is a regular octagon.

An isosceles triangle is one with two sides equal. An isosceles triangle has a line of symmetry passing through the middle of the angle formed by the two equal sides. If the triangle is cut out and folded in half along this line of symmetry the two angles opposite the equal sides match each other. In this way we can discover practically that a triangle with two equal sides always has two equal angles.

Finally, a scalene triangle is one with no equal sides. Using these different categorisations it is then possible to determine seven different kinds of triangle.