Cut an Illustrator circle in half ? With the most recent versions, you can now simply create an ellipse using the ellipse tool (hold down the shift to get the required proportions) and with the live artwork selected go to the properties panel and click the first set of '...' to see more options and then go to the pie start angle field and enter 0 and then go to the pie end angle field and enter 180 and there you have your semi design in seconds. Not sure why the pie slice angles are not on the properties panel proper but instead require the additional action of clicking the more options.
It is live so if you want to use it as a standard artwork, go to the object menu and shape and expand command. Once you have expand the design into a regular shape then you can't use the features associated with the tool but you can now access the points using some of the tools in the application
You can use this but you can also use the curvature etc as well to manipulate the artwork. You don't have to but you can distort the design or add additional points etc to the artwork
You can alway save the two halves to your library so it is always accessible. Of course, once you have your artwork you can add it also to the symbols panel or perhaps use it with the repeat feature and go to the object menu and repeat and radial. You can continue to work with the semi design, perhaps add a shadow to the design etc Or perhaps use it with the blend tool and then save the result of any of this work to the symbols panel as well as the libraries panel
This one uses the powerful divide feature found in the pathfinder. Select the ellipse tool. Draw the artwork with the ellipse tool by holding shift key down. View menu and smart guides ON. Draw a rectangle using the rectangle tool bigger than the circle. Shift the rectangle to cut the middle of circle (intersect will appear in the smart guide). Pathfinder panel and divide. Object menu and ungroup. Delete remaining rectangle. Once you have divide the path, you still need to ungroup via the object menu and then you can remove the other parts of the path to leave the cut circle
Create your shape using the ellipse tool (and hold down the shift) or select an existing path. Go to the eraser tool. Select the knife tool (beneath eraser tool). Hover over top anchor point and hold alt / option. Draw through the top anchor. Through to the center to the bottom anchor point. The alt / option key makes the knife a straight line knife. Once you have cut through your artwork, you can remove the other half
You can also use the eraser tool, requires a quick set up of the eraser by double clicking the eraser tool and setting the angle to 90 and roundness to 0 etc and the size doesn't particularly matter though if you want set it to the max. Hover over the top point in the design and click with your eraser tool (if max and bigger than the circle) or draw down from the top to the bottom with the eraser tool and holding down the shift key so it is a straight line cut