# Unit Circle Chart

The unit circle chart shows the position of the points along the unit circle that are formed by dividing the circle into eight and twelve parts. The coordinates of each point can be solved for using the one of the two corresponding special triangles.

## Building the Chart

Draw a circle of radius one at the center of the cartesian coordinate plane. Make sure to draw it big enough to leave space for all of the points or print out a blank unit circle chart. Label the four points where the circle intersects the and axis.

### Divide Unit Circle by Eight

Divide the circle into eight equal parts and label the angle corresponding with each point. On the left, the angles are measured using radians, where one full rotation is equal to . On the right, the angles are measured using degrees, where one full rotation is . By convention, angles are measured from the right-most edge of the circle and the positive direction is counter-clockwise.  To find the position of the first point, start by observing that the 45-45-90° special triangle and the coordinates of the first point at the angle () are similar triangles.  Use the properties of similar triangles to solve for the and sides of the of the triangle formed by the point.

This gives us the position of the first point along the unit circle. Note, mathematicians prefer not to have a radical in the denominator and so they rewrite the fraction: . Now that we have found the position of one point, we can use symmetry to fill in the other three points that correspond with the 45-45-90 triangle. This is the same as reflecting the horizontally over the -axis, vertically over the -axis, and diagonally over both axes.   Filling in the these points gives us the rest of the positions formed from dividing the unit circle into eight parts.

### Divide Unit Circle by Twelve

Now, we can repeat the same process for the next set of points. Start by dividing the circle into twelve equal parts and label the angles corresponding with each point.  Observe that the 30-60-90° triangle and the triangle formed by the coordinates of the first point at (30°) are similar triangles.  Use the properties of similar triangles to solve for the x and y component of the point.

This gives us the coordinates of first point at radians (). Then we can take advantage of the symmetry within the first quadrant of the coordinate plane to find the position of the next point associated with the 30-60-90° triangle. This is visualized by reflecting the triangle diagonally. Like before, we can reflect the position of a point over the axes to find the other points corresponding to that point. Follow this process for each of the two points corresponding to the 30-60-90° triangle in the first quadrant:   Filling in these points gives us all the positions of the points formed by dividing the unit circle by as shown in the figure below.