Measure the diameter. (Use unused, unminted dental floss, or food grade string. You get the idea.)

The radius is one-half the diameter.

From the center of the circle (the top of the cake, and yes, I know, this is done by hand, and will not be perfect), drag an imaginary “chord” (look this up: it’s a line segment shorter than the diameter, that goes from one point of the circle to another) parallel to a chosen diameter until you are half a radius length from the center.

From one endpoint of the chord to the radius to the other endpoint of the chord is 120 degrees, or one-third of the circle.

Cut this wedge out.

Cut the remaining cake in half.

DONE.

You can prove this by cutting a circle into four quadrants, like they taught you in high school. Create an arc from zero degrees to just past 90 degrees, and another arc from zero degrees to just before 270 degrees. The smaller arc created you would want to be 120 degrees = one-third of the circle. If you put two right triangles back to back, sharing the center of the circle as a point on one edge of each triangle, you create two right triangles which share an edge along the radius that is pointing at 180 degrees. The angle created at each corner of the triangle which is sharing the center point of the circle is 60 degrees.

60 plus 60 = 120 degrees, is an arc carving out a third of the circle’s area.

The right triangle that is 30-60-90 degrees has sides 1/2 – sqrt(3)/2 – 1, for a unit circle. Multiply by radius “r” for any other length.

The side that is 1/2 is the side opposite the 30-degree angle. 1/2 the RADIUS.

So grab a string that is the length of the diameter. Double it to get half the diameter, or the radius. Mark out where along the 180-degree line where 1/2-radius is. Then create a chord parallel to the vertical diameter (90 degrees to 270 degrees), such that this chord intersects this point. Where the chord touches the circle are your two angle endpoints, and the center is the corner point of a 120-degree angle.

A cake cut into three equal parts.

Enjoy!!!!!

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