The Quadrature of The Parabola - Geometric Proof - Areas of The Triangles

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

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