The geometry of quadrilaterals

​ ​The quadrangle: it has only one more side than the triangle, yet it is completely different. For example, there is no equivalent of the Pythagorean theorem for quadrangles. And if you know all the sides of a triangle, then the triangle is unique. But a quadrangle can change its shape without the sides changing their lengths. This is the reason we use triangular structures in construction of the objects that have to carry a load, like bridges. ​ ​Look at bridges and their beam trusses on scaffolding or at the back of a bookshelf and you'll find places where they put a diagonal on top of a rectangle, so that it forms two triangles.

Try it yourself, and you can feel how much stronger the structure gets. ​ ​But we're going to talk about quadrangles, not triangles. What all quadrangles have in common is, of course, that all of them have four angles. Thus, they also have four sides. This one looks a bit crooked. If we make all the angles equal exactly 90 degrees, in other words, make them all right angles, we get a rectangle.

A rectangle is a quadrangle with only right angles. And if all the corners are all right angles, then every side automatically becomes as long as the opposite side. ​ ​A special case of a rectangle is a square. A square is a quadrangle that has only right angles, and all four sides of which are of equal length. If we push a square a bit from the side like this, it's no longer a square. Because the angles are not 90 degrees anymore, now it's a rhombus.

A rhombus is a quadrangle in which all the sides are of the same length, and the opposite angles are equal. You mostly see rhombuses in this orientation. Diamonds in card games and the logo of the car maker Renault are examples of rhombuses. ​ ​If we expand a rhombus like this, then all sides are no longer equal. We now have a parallelogram. In a parallelogram, the opposite sides are equal.

Just like a square is a special case of a rectangle, a rhombus is a special case of a parallelogram. If we straighten a parallelogram like this, we get a rectangle again, and if we do like this, so that the opposite sides are no longer equal, we get a trapezoid. Now two opposite sides are parallel while two others are not. If we make the sides that are not parallel equal, we get an isosceles trapezoid. ​ ​Here are some quadrangles to start with: a rectangle, a square, a parallelogram, a rhombus and a trapezoid. There are more, but we'll settle with these for now.