Area of a circle, analyzing with trigonometry and geometry

I was trying to find the area of a circle decomposing it into polygons. So looking at the parametric equations of a circumference and the formula to define the area of a polygon having its coordinates i did the following calculus:

(I) Parametric equations of circle:

x= r * cos(t) I assume r=1 so I must get: **x= cos(t) **and **area=pi**

y= r * sin(t) **y= sin(t)**

(II) Area of a polygon having its coordinates:

** A(poly)=1/2 * | Xa Xb Xc Xd ... Xn Xa|**

| Ya Yb Yc Yd ... Yn Ya|

Giving any value of 't' i will get a point of the circle, so i imagined a polygon of 360 sides (I used t as degrees) that would have an area very near the area of the circle.

** A(circle)=1/2*| cos(1) cos(2) cos(3) ... cos(360) cos(1) |**

| sin(1) sin(2) sin(3) ... sin(360) sin(1) |

Developing the determinant I get:

A(circle)= 1/2* | [cos(1)*sin(2)+cos(2)*sin(3)+...+cos(360)*sin(1)]-

[sin(1)*cos(2)+sin(2)cos(3)+...+sin(360)*cos(1)] |

And finally syntesis of the formula to get circle's area:

**1/2***

{summation of t=1 to t=359} cos(t)* sin(t+1)- sin(t)* cos (t+ 1)

+cos(360)*sin(1)-sin(360)*cos(1)

Using the calculator On-Line Calculator to find the result of the sum a got a very big number and after some adjusts i got still a number very far away from pi. Could someone tell me what did i make wrong?

Sorry by mistakes in grammar, i am not native.