Carpentry and Woodworking - How do I construct a round tabletop extension?

I have an antique oak round table measuring 36" in diameter, with no leaf. It's just not big enough for dinners for company so wd like to make a 'topper' out of plywood to extend it to about 48" in diam. Was thinking for storage ease, to make it fold in half by perhaps using a piano hinge. Has anyone done anything like this or have any ideas/input? And, what equipment wd I need to measure an even circle? Any help wd be appreciated.

Working from the bottom side of the piece of plywood:

This will only take a half-sheet of plywood, so measure 48" down the 8' length of the 4'x8' sheet of plywood. Cut the sheet in half producing a square piece of plywood 48" square.

Snap a chalk line across both diagonals of the square. Where they cross is the center of the circle. Using a short nail or screw inserted into the center, tie a piece of string around the nail/screw. Using a pencil placed at the closest point on any of the sides, tie the other end of the string around the pencil point.

Holding the pencil vertical, mark a circular line completely around the center point returning to the starting place. You may need to shorten the string to avoid falling off the surface. Do not let the string wind up on the center post, otherwise a spiral is marked rather than a circle.

Continuing to work with the wood upside down, use a jigsaw to cut just outside the marked line. When nothing but a round disk remains, sand the perimeter to a smooth finish. Edge-band if desired.

Easiest way to cut it would be with a router , you can buy or even make a router trammel yourself. perfect circles every time.

Just adding to IBM5081's excellent directions:

A stick with a nail in it makes a slightly more accurate trammel. It doesn't stretch as string may. If you slightly cut into it with a knife, the knife will like to ride there and can mark sharper than a pencil.

I recommend tracing two circles: one for your cut, and one about 1/8" inside it for a visual aid in trueing the circle after using the jigsaw. A router on trammel saves you these efforts.

Nail three or four little blocks to the underside, to keep this top from siding around on the real tabletop.

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If you want to half the size stowed, here's one idea: before marking the circle, jigsaw a puzzle joint (as in jigsaw puzzle, zig-zag, or even some meaningful pattern) near the midpoint. Any odd cut keeps the leaves aligned so long as they're drawn together.

Use four screw-in eyehooks under the plywood, with bungy cords, to draw tight in use. The cords may also bind the leaves together in storage.

Specialty hardware meant for locking table leaves together looks better. Leevalley.com has some.

If you really want a hinge there are a few nice ways that could be done, just ask.

I'm really amazed at the response regarding instructions to make an extension for my tabletop. Was my first time to post, so it was gratifying to receive...I appreciate all the advice to get me started on this project. Thanks a lot. :D