# Happy Pi Day 2018

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.

- 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