Unit Circle Chart

This chart represents all of the points formed by the two special triangles with a hypotenuse of one. The two special triangles are often referred to by their angles: 30-60-90 and 45-45-90 degrees. The triangles are special because their angles perfectly divide the unit circle into equal parts.

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 respectively.
unit circle chart.pdf
unit circle chart (blank).pdf

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.

A circle of radius one drawn at the center of the cartesian coordinate plane.

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.

The angles that represent dividing the unit circle by eight.

Observe that the 45-45-90 degree triangle and the first point at π/4 are similar triangles.

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

This gives us the point at 45 degrees or π/4 radians.

We can use symmetry to fill in the other three points that correspond with the 45-45-90 degree 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 degrees. Divide the circle into twelve equal parts. Each section has an angle of 30 ° or in radians, π/6 rads.

Observe that the 30-60-90 degree triangle and the triangle formed by the first point at π/6 are similar triangles.

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

This gives us the first point at π/6 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 observe the same symmetry as before for both for both points π/6 and π/3.

We now have finished constructing the 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 respectively.
  1. Understanding the Unit Circle
  2. Unit Circle Chart (pdf)
  3. Unit Circle Chart Blank (pdf)
  4. Radians
  5. Pythagorean Theorem