# Unit Circle Chart

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

## Building the Chart

Draw a circle of radius one at the center of the cartesian coordinate plane. Label the four points where the circle intersects the x and y axis.

We are going to start by finding the points corresponding with the 45-45-90 degree special triangle. Divide the circle into eight equal parts, and label the corresponding angles.

Observe that the 45-45-90° special triangle and the coordinates of the point at the angle (45 °) are similar triangles.

Use the properties of similar triangles to solve for the x and y component of the coordinate.

This gives us the position of the point at 45 degrees or radians.

We can use symmetry to fill in the other three points that correspond with the 45-45-90° triangle.

This gives us roughly half of the chart.

Now that we are done with the first special triangle, we can repeat the same process for the next special triangle of 30-60-90°. Divide the circle into twelve equal parts. Label each point with its corresponding angle. Note that the angle between each point has an angle of 30° or radians.

Observe that the 30-60-90° triangle and the triangle formed by the coordinates of the first point at are similar triangles.

Use the properties of similar triangles to solve for the x and y component of the point.

This gives us the first point at radians.

Then we can take advantage of the symmetry within the first quadrant of the coordinate plane, which is the same triangle reflected diagonally.

Then we can use the same symmetry as before for both to fill in the positions of the other ten points.

We now have finished constructing the unit circle chart.

## Download PDF

## Examples

### Memorize Unit Circle Chart

To memorize the unit circle chart, you can learn the positions of the points at the angles 30 degrees and 45 degrees, then use symmetry to fill in the rest of the points.