Polygons and PolyhedraPolygons
A A polygon is geometric shape that is made up of straight line segments. Polygons cannot contain any curved sides, or holes. For example, a square is a polygon but a circle is not.
We give different names to polygons, depending on how many sides they have:
Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Heptagon
7 sides
Octagon
8 sides
Angles in Polygons
Every polygon with n sides also has n The internal angles of a polygon are the angles on the inside, at every vertex.
° + ° + ° + ° =
° + ° + ° + ° + ° =
It looks like the sum of internal angles in a quadrilateral is always
The same also works for larger polygons. We can split a pentagon into
A polygon with
Sum of internal angles in an n-gon
Convex and Concave Polygons
We say that a polygon is A concave polygon has at least one internal angle greater than 180°. At least one of the diagonals lies outside the polygon. A common way to identify a concave polygon is to look for a “caved-in” side of the polygon. Concave is the opposite of convex polygons. A convex polygon contains no internal angles greater than 180°. All diagonals lie inside the polygon. This is the opposite of concave polygons.
There are two ways you can easily identify concave polygons: they have at least one internal angle that is bigger than 180°. They also have at least one diagonal that lies outside the polygon.
In convex polygons, on the other hand, all internal angles are less than
Which of these polygons are concave?
Regular Polygons
We say that a polygon is A regular polygon is a polygon in which all sides have the same length and all interior angles have the same size.
Regular polygons can come in many different sizes – but all regular polygons with the same number of sides
We already know the sum of all The internal angles of a polygon are the angles on the inside, at every vertex.
If
The Area of Regular Polygons
Here you can see a A regular polygon is a polygon in which all sides have the same length and all interior angles have the same size.
First, we can split the polygon into congruent,
We already know the
Notice that there is a right angled triangle formed by the apothem and half the base of the isosceles triangle. This means that we can use trigonometry!
The base angles of the isosceles triangle (let’s call them α) are
To find the apothem, we can use the definition of the
Now, the area of the isosceles triangle is
The polygon consists of of these isosceles triangles, all of which have the same area. Therefore, the total area of the polygon is