**Areas of The Triangles**

In propositions eighteen through twenty-one, Archimedes proves that the area of each green triangle is one eighth of the area of the blue triangle. From a modern point of view, this is because the green triangle has half the width and a fourth of the height:

By extension, each of the yellow triangles has one eighth the area of a green triangle, each of the red triangles has one eighth the area of a yellow triangle, and so on. Using the method of exhaustion, it follows that the total area of the parabolic segment is given by

Here *T* represents the area of the large blue triangle, the second term represents the total area of the two green triangles, the third term represents the total area of the four yellow triangles, and so forth. This simplifies to give

Read more about this topic: The Quadrature Of The Parabola, Geometric Proof

### Famous quotes containing the words areas of, triangles and/or areas:

“The ambiguous, gray *areas of* authority and responsibility between parents and teachers exacerbate the distrust between them. The distrust is further complicated by the fact that it is rarely articulated, but usually remains smoldering and silent.”

—Sara Lawrence Lightfoot (20th century)

“If *triangles* had a god, they would give him three sides.”

—Charles Louis de Secondat Montesquieu (1689–1755)

“The ambiguous, gray *areas* of authority and responsibility between parents and teachers exacerbate the distrust between them. The distrust is further complicated by the fact that it is rarely articulated, but usually remains smoldering and silent.”

—Sara Lawrence Lightfoot (20th century)