Convex, concave and self-intersecting polygons

From Graphic maths
Level: GCSE / A Level Maths

Many of the polygons you meet in maths are convex, like this one:

Convex-poly.png

There are several other types of polygon, including concave and self-intersecting polygons, which we will meet here.

Concave polygons

Here is an example of a concave polygon:

Concave-poly.png

A concave polygon has one or more internal angles that are greater than 180 degrees. In the example above the two marked angles are both greater than 180 degrees. This means that some of the vertices of the shape point inwards towards the centre. In a convex polygon, all the vertices point outwards.

Properties of concave polygons

For a concave polygon:

  • It is possible to draw a diagonal of a concave polygon that is outside the shape itself. This is shown as the green line in the diagram below.
  • It is possible to draw a straight line through the polygon that crosses more than two sides of the polygon. This is shown by the red line.

Concave-properties.png

The area and perimeter and internal and external angles of a concave polygon are calculated in the same way as for an irregular convex polygon.

Concave polygons can have any degree of line or rotational symmetry, just like other irregular polygons.

Convex hull of a polygon

If you were to place an imaginary elastic band around a concave polygon, it would form a convex polygon that completely encloses the original polygon:

Convex-hull-poly.png

This shape is called the convex hull.

The convex hull of a convex polygon is the same as the original polygon


Self intersecting polygons

Convex and concave polygons, like the ones above, are called simple polygons

A self intersecting polygon has sides that cross over. These are not classes as simple polygons (they are sometimes called complex polygons).

Here are two examples:

Pentagram.png

The first is a crossed rectangle. This is like a normal rectangle, except that the top and bottom sides cross over in the middle.

The crossed rectangle only has 4 vertices. The point in the middle where two sides cross is not an extra vertex


The second shape is a pentagram. This shape has 5 vertices, the 5 tips of the star shape. It is like a pentagon, but the vertices are connected differently. Once again, the points where the lines cross are not counted as extra vertices

The rules for calculating the angles of a self intersecting polygon are different to a simple polygon. We won't cover them here.

The method of calculating the area and perimeter is also different, in part because it is not obvious which areas are inside or outside the shape. Here are two possible interpretations of the inside of a pentagram:

Pentagram-area.png

Even when you have decided how to define the inside of the shape, the actual calculation requires you to calculate where the sides cross. Again this is not covered here.

See also