Given that you have a piece of circular material of some diameter. There are no markings on it. Is there any easy and practical way to determine the center point and draw two lines (one perpendicular to the other) through the center so that I can section the circle into equal quads?
Stupid, simple answer first: Can you fold this material? Makes it kind of trivial…
If you’re looking for a “compass and straight edge” kind of answer, mark a point on the circumferance, then two points equidistant from that point intersecting the circle. Connect those two points and construct a perpendicular on that line through the original point. That second line is a diameter of the circle. Bisect that diameter (obviously that’s going to be at the center of the circle) with another line to get a second diameter splitting the circle into quads.
Although that second method requires having material outside the circle itself to do all the constructions.
If you’re dealing with a more physical application and you can’t fold it (say a circle made of metal) fix a string to one point of the circle, keep it taut and use that to make a straight line following along the circumferance, when you start to get slack, back up a bit. There’s one diameter. Use a ruler to find the half-way point, and a protractor to get a 90[sup]o[/sup] angle to make the second.
Do any of those work? What kind of limitations are we dealing with? What’s the application? And tell me I didn’t just help you with your homework…
There is a way, but I can’t remember it at the moment. I know we covered it back in geometry when we studied constructions - given a length of arc, find the center of the circle.
Ok, now I remember. Place a compass point on the edge of the circle, and make two marks intersecting it (the edge). Draw a line segment connecting these two points. Find the perpendicular bisector of this line segment. This new line passes through the center of the circle.
To find the exact center point, and to divide your circle into four equal parts, find the perpendicular bisector of the diameter segment you found above.
Practical? Well, if you’re dealing with a piece of paper, yeah. If you’re trying to slice a cake fairly, it may be a bit tougher.
Seven minutes too slow…
I’ll have to think about them. This isn’t for homework 
but for a prototype development project I am working on. The circle is an acrylic disk, so it can’t be folded.
How do I determine the two points that are equidistant from the first?
I don’t understand this one at all. Not clear about “make a straight line following along the circumference”. Text with this kind of stuff is a bit limiting as I’m more of a visual person, sigh.
put the point of the compass on the first mark, then draw two marks - one on
either side.
(Although, surely any perpendicular from the center of a chord will pass
through the center of the circle ? So all you would need to do is draw any
chord and bisect it )
Euclid III.1
Don’t they teach you anything in school anymore?
If this is a real problem with actual bits of acrylic, rather than a math problem, then you just need a centre square and a set square. With those two tools and a pencil or a scribe, you can mark your quadrants in under a second.
This my first post ever, so I might find myself in the bbq pit soon…We’ll see.
I’d like to suggest a similar solution that only requires a compass and a ruler, and will hopefully be quite clear. It’ s a bit longer obviously. Here goes :
1/Imagine the circle is a clock. Pin the compass point some place on the edge, that we’ll call 12.00.
2/Stretch the compass to mark a second point somewhere near 4.00 (approx.). that’s “A”
3/ Leave the compass pinned on 12.00, and with the same compass length, make a mark somewhere near 8.00. That’s “B”.
4/ Leave the compass pinned on 12.00, Make a mark near 10.00 or 11.00 ( “C” ), then with the same compass length, somewhere near 1.00/2.00 ( “D” )
5/Join A,B,C,D. That’s some sort of a rectangle. Trace the diagonals of the rectangle.The point where the diagonals meet is the center of the circle.
6/Trace a line from 12.00, through the center, to 6.00. That’s your first line to cut the cake…
7/Measure the segment [BC], to find its middle. Do the same on [AD]. Trace a line between the two points which is the second line you’re looking for.
8/There is a way to use the compass instead of measuring, but I think I’ve gone on long enough with this…
9/There must a simple clever way to go about the whole enterprise, but where’ s the fun.
er … that won’t work ! Unless you’re very lucky !
and sciguy’s method works with just a compass & ruler !
Acrylic disks… heck, why didn’t you say so. I can solve your problem without the need for any compass, rulers, or straight edges.
First, put a piece of paper over the acrylic disk. Draw a circle on the paper at the perimeter of the disk. Cut the paper if desired. Fold the paper in half so the circle overlaps. This gives you a semi-circle. Now fold the paper in half the other way, so again the lines overlap. This gives you a four-sheet thick quarter-circle wedge.
Unfold and you now have a template mask for the cut you need to make.
Tape the paper back over the disk and make your cuts (or mark the disk).
I’ll address the first question. To find the exact center of any flat object (of uniform material), regardless of shape, get a small nail, a piece of string, a small weight and a pencil.
Put the object up against a wall and lightly hammer the nail through it into the wall. Let the object swing back and forth until it finds equilibrium. Wrap the string around the nail, tie the weight to it, and let it dangle as well. Use the pencil the trace the string’s path along the object.
Now, remove the nail, rotate the object any amount and repeat.
The point where the two lines cross is the center of the object and is the point where you could balance the object on the end of a pencil.
If you don’t want to pound nails into your object or the walls, you could just pinch the object and string between your fingers and let them swing.
This seems like the easiest approach. Sometimes it is difficult to see the forest through the trees. 
Thanks everybody for all your ideas!
Glad to have helped! Nyah nyah, he picked my answer. I win, hah! 