A circle is the set of points that are a given distance r from a given point p.
Here p is the center and r is the radius.
The unit circle has radius 1, and is centered at the origin,
if your plane has an origin.
The distance across the circle is the diameter, denoted d, and is equal to 2r. The circumference, denoted c, is the distance around the circle.
We're not going to delve into arc length here, as that requires advanced calculus. However, the ancient Greeks could certainly wrap a string around a circle and then measure its length. They knew that the circumference was about 3.14 times the diameter, and they knew that this ratio wasn't exact. It never seemed to "come out even". So they assigned it the Greek letter π. Thus the circumference is always πd, or 2πr. We now know that π = 3.1415926535…
With this in hand, they quickly proved that the area was πr2. Their proof is as beautiful today as it was 2,000 years ago, and it only takes a little tweaking with limits to make it rigorous. It is based on the axioms of area. If a circle is cut into n wedges, each 360/n degrees, each piece has the same area. After all, the pieces are identical; they are just in different locations in the plane. So cut a circle of radius r into ever smaller wedges as follows.
Take the bottom half of the unit circle and cut it into many wedges, like thin slices of pie. Place these wedges next to each other, points up, like the bottom teeth of a wild animal. Split the top half of the circle into wedges pointing down, the top teeth of the animal. Now close the animal's mouth. The teeth interlock perfectly. As the number of teeth increases the shape approaches a rectangle. The width is πr (half the circumference), and the height is r. Therefore the area is πr2.
In 300 BC, Archimedes developed a proof for the volume of a sphere that didn't require integral calculus. However, recent discoveries suggest he practically invented calculus onhis own, 2,000 years before Newton and Leibnitz. So it's hard to say that his proof does or does not entail calculus. (Personally I think it does.) If it isn't calculus per se, it's something awefully close. Here is a rough outline of his approach.
Place an inverted cone next to a hemisphere of the same dimensions, and show that the two cross-sections, at any height, add up to the area of the base circle, or πr2. The hemisphere and cone together fill a cylinder of volume πr3, and he knew, using other methods, that the volume of the cone was 1/3 that of its containing cylinder. That left 2/3 πr3 for the hemisphere, or 4/3 πr3 for the sphere. You can learn more about Archimedes and his circles at pbs.org.