USING TRIG TO DETERMINE PI
One way to determine the circumference of a circle is to find the perimeter of
a regular (all sides equal) polygon inscribed within the circle. The more sides
the polygon has, the closer its perimeter is to the circumference of the circle.
Then if you let r = 0.5, you can get a very good estimate for the value of pi.
In the diagram above, a hexagon is inscribed in a circle.
The hexagon is divided into _______ triangles, each identical to ΔAOB.
In ΔAOB, let p represent the length AB, r the radius OB and θ the angle AOB.
Write a trig equation relating p,r and θ : ___________________ ….. (1)
Rearrange (1) to solve for p :
___________________ ….. (2)
Write an equation for the perimeter, P, of the hexagon in terms of p :
_______________ ….. (3)
Replace p in (3) by the expression in (2) to get a formula for P in terms of r
and θ : _____________ …. (4)
One complete rotation is how many degrees ? __________ In a regular
hexagon, what is the measure of angle θ ? _________
Suppose r = 0.5 units. Calculate using (4) the perimeter of the hexagon : P =
____________.
Calculate the circumference ( C= 2pr) of the circle in terms of p : C =
___________.
By increasing the number of sides in the polygon, the perimeter should get
closer and closer to the circumference of the circle.
In order to do this, we need to adjust formulas (3) and (4) for a polygon of n
sides.
Formula (3) becomes __________ …(6) . Formula (4) becomes
______________ …(7)
What is a general expression for calculating θ in terms of n and 360° ?
_____________ ….. (8)
Finally, substitute for θ from (8) into (7) to get a general formula for the
perimeter : P = __________________ … (9)
Now, with the aid of a calculator, use formula (9) to complete the following
table. Round P correct to 8 decimals.
n
6
60
600
6000
600000
6000000
60000000
600000000
6000000000
60000000000
600000000000
P
CONCLUSION
THE VALUE OF PI CORRECT TO 8 DECIMAL PLACES IS
_________________________________ .