On the Measurement of the Circle

Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis) is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work.

Proposition one states:

The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.

Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion.

Proposition two states:

The area of a circle is to the square on its diameter as 11 to 14.

This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition.

Proposition three states:

The ratio of the circumference of any circle to its diameter is greater than ${\displaystyle 3{\tfrac {10}{71}}}$ but less than ${\displaystyle 3{\tfrac {1}{7}}}$.

...
Wikipedia