Archimedes proved a circle's area equals a triangle's by ruling out alternatives
Unroll a circle's edge into a straight line, stand the radius upright at one end, and join the tips. Around 260 BCE Archimedes showed that this right triangle has exactly the same area as the circle. Half the circumference times the radius gives the familiar formula: pi times the radius squared.
The Greeks crept up on this result in stages. Hippocrates of Chios first showed that a disk's area grows with the square of its diameter, while working on his famous lune, though he could not pin down the constant involved. Eudoxus of Cnidus reached a similar proportionality with the radius. Archimedes, in his Measurement of a Circle, supplied the exact answer.
His argument is a double proof by contradiction. Suppose the circle were larger than the triangle by some excess. Inscribe a square, then an octagon, then polygons with ever more sides, until the slivers left between polygon and circle total less than that excess. The polygon would then have to beat the triangle, yet its perimeter is shorter than the circumference and its inner height shorter than the radius, so it cannot. Now suppose the circle were smaller; circumscribed polygons squeeze in from outside and produce the mirror-image contradiction. Only equality survives. This method of exhaustion was one of the earliest uses of what we now call a limit, although modern mathematicians note that it treats some comparisons of curved and straight lengths as obvious.
A more visual route was used by Nicholas of Cusa, Leonardo da Vinci and the Japanese mathematician Satō Moshun. Slice an inscribed hexagon into triangles from the centre and rearrange them into a parallelogram. With more and more sides, that shape approaches a rectangle whose width is half the circumference and whose height is the radius, which again multiplies out to pi times r squared.
Strictly, a circle is only its boundary curve, which covers no area at all; the region inside is a disk. Mathematicians therefore speak of the area of a disk, while everyone else keeps saying circle. Pi itself, the ratio of circumference to diameter, is roughly 3.14159.
Source: Area of a circle
