Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Measurement of a Circle Measurement of a Circle Measurement of a Circle or Dimension of the Circle (Greek: , "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