A **polygon** is a plane shape (2D) with straight lines. It consists of **vertices** and **edges**.

A polygon is **regular** when all angles are equal and all sides are equal. For instance a regular pentagon consists of 5 vertices and 5 edges of equal size. The vertices of a regular pentagon are equally spread on a circle. This outside circle is called a **circumcircle**, and it connects all vertices (corner points) of the polygon. The radius of the circumcircle is also the **radius of the polygon**. We can use the **trigonometric formulas** to work out the **(x,y) coordinates** of each vertex of a regular pentagon. (See picture on the right).

Using this approach, we can use a Python script to calculate the (x,y) coordinates of the 5 vertices of a regular pentagon and store them in a list of [x,y] sub-lists.

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pentagon=[] R = 150 for n in range(0,5): x = R*math.cos(math.radians(90+n*72)) y = R*math.sin(math.radians(90+n*72)) pentagon.append([x,y]) |

#### Star Shapes

A

**pentagram**is a polygon that looks like a 5-pointed star. The outer vertices (points of the stars) form a regular pentagon. The inner vertices of the star also form a smaller “inner” regular pentagon.

We can hence use a similar approach to calculate the (x,y) coordinates of both “outer” and “inner” vertices of our pentagram.

#### Python Turtle

We have completed the code to calculate the coordinates of a regular pentagon (using the code provided above) and have created a function called

*drawPolygon()*that uses Python Turtle to draw a polygon on screen.

#### Your Task

Adapt the above Python code to calculate the coordinates of all the vertices of a pentagram (star shape) and draw the pentagram on screen.

The output of your program should be as follows:

#### Extension Task

Use the shoelace algorithm to calculate the area of your pentagram.