# Investigation of circumference ratio - finding the value of pi.

Stoney Li 11g HL math

Explain why the marked angle is 30

In this diagram, the C point is the center of the circle and hexagon. The triangle ACD is one sixth of the hexagon. So the angle ACD is one sixth of central angle. The central angle is 360 degree, which mean the angle ACD is 60 degree.  The segment BC connects the midpoint of segment AB to the point C. So segment AB equal segment BD. Because the segment AC and segment CD are radius of circle, the triangle ACD is isosceles triangle. That mean segment BC divides the triangle ACD to two triangles and this two triangles are equal. Which mean angle ACD had divided to angle ACB and angle BCD. So 60-degree divide by 2, the angle ACB equal angle BCD equal 30 degree.

Use trigonometric ratio(s) to find the area of the hexagon.

At first we find the area of triangle ACD.

Known:

Segment AC=1

Segment BC=1

Angle ACD=60

We know two sides and one included angle. Then we could use the formula to find the area of triangle.

As we known, the area of triangle ACD is one sixth of area of hexagon.

Let us try 12-sided, 24-sided, and 48-sided polygons.

12-sided:

The triangle ACD in 12-sided is one twelfth of the .

Known:

Segment AC=1

Segment BC=1

Angle ACD=30

Area of triangle ACD:

Area of :

24-sided:

The triangle ACD in 24-sided is one twenty-fourth of the .

Known:

Segment AC=1

Segment BC=1

Angle ACD=15

Area of triangle ...