# Pi Day 2013

Happy Pi Day 2013, this interactive activity will guide you through ways to find π using polygons.

Most people know the definition of PI as the ratio of the
circumference of a circle to its diameter. They have probably used PI
in circle problems, calculating the area using
```
area = πr
```^{2}

or the circumference using
`C = 2πr`

or
`C = πd`

. But what of this mysterious number, is it possible to create a
visual analogy. This demonstration attempts to do so by examining and
manipulating a regular polygon.

This will be done in two ways firstly by considering the
perimeter of a regular polygon which fits perfectly in a circle of
radius one, the polygon can be opened to form a straight line and its
perimeter then measured. To show how π relates to area we again use
a polygon, dividing its area into triangles.

## The controls

For the sake of consistency with the other activites the
controls are described below. But the best way to use this activity is
to follow the instructions guide in the top right hand corner

- Next button is pressed to move to
the next instruction
- Previous button is pressed to move
back to the previous instruction
- Focus select the part to focus
and zoom in on
- Area toggle area or
circumference
- Exterior angles toggle to
display exterior angles
- close polygon click button to
animate the polygon closing
- open polygon click button to
animate opening the polygon
- Exterior Angle Adjust this slider
to change the exterior angle which in effect opens or closes the
original polygon
- side length Adjust this slider to
change the side length
- side length Adjust this slider to
change the number of sides.
- Automatic recalculate - when
this button is on changing the number of sides and the two other
sliders will automatically adjust the angle and length to form a
regular polygon that fits perfectly in the circle.

## Related Activities

If you like visual mathematics why not check out the interactive multiplication tables
The inspiration for this activity actually came from playing around
with the interactive fractal after
setting the branches to one. The interactive
polygon explorer. is better for teaching the various properties
and angles of polygons.

You can follow the instructions here to guide you through the
activity. Or just play around and discover for yourself. Words
in blue refer to interface controls, click them to be directed
to the control.

Click next to continue.

Let's begin by focusing on the circle. Inside is an equilateral triangle, a regular
polygon of three sides

- The circle has a radius of one
- Each vertex of the polygon lies on the circle

Notice the exterior angles matches the angle
slider value.

In automatic mode when you change the number of sides, the
angle and side length are recalculated to maintain a regular polygon

Drag the sides slider to see

Setting the sides to six gives a hexagon, notice it has nice
numbers

- The exterior angle is 60°
- The side length is 1
- This equals the circle radius of 1
- Can you explain why?

Now we are going to make some changes to the angle
slider

Drag the slider, notice what happens to the polygon as the angle approaches zero.
As well as using the slider you can click the >
button to open the polygon, or < to
close it.

So opening up a polygon allows us to use the ruler to measure
its perimeter. For the hexagon the perimeter is six since there are
six sides each of length one.

We have hidden the exterior angle, to show it again click the exterior angle button.

With twenty sides, the polygon start to resemble another shape. Open up the polygon and measure its perimeter.

Now increase the sides and observe what the
happens to the perimeter.

At 100 sides our polygon is now a close approximation of a
circle. The perimeter of the polygon will therefore be an
approximation of the circumference of the circle.

The formula for the Circumference is

C=2πr

r in this case is 1 the perimeter=2π So our ruler value
gives us the value of 2π

What about the area of a circle?

We can consider the polygon to be made up of lots of isosceles
triangles.

Click the area button so it is on

We can do the same trick and unfold the polygon

Move the angle slider to zero

We could work out the area by adding up the area of the
triangles

But lets try something else

Press the > button to animate.

After the animation we have a new shape, a parallelogram.

With 11 sides an odd number, the end result will be a
trapezium.

But we will concentrate on the area of a parallelogram produced
with an even number of sides.

The area of a parallelogram is equal to the its base length x height

To calculate the height we need trigonometry, but instead try
increasing the number of sides and examine what happens to the
shape.

With 100 sides the height of the parallelogram is approximately
the radius of the circle or 1

So the area of the circle is equal to

1 x base-length.

1xbase-length = πr^{2}

r=1 so base= π

So the ruler measurement gives us the base-length which is π

# The End

## Happy Pi Day 2013