A Toulmin Analysis of the Pythagorean Theorem: Geometric Proof Through Square Area Equivalence

Overview

This paper presents a Toulmin logical analysis framework applied to a geometric proof of the Pythagorean theorem. The analysis demonstrates how the theorem A² + B² = C² is proven by constructing nested squares and comparing their areas through five systematic steps, ultimately establishing that the relationship holds for all right-angled triangles.

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Toulmin Analysis of Pythagoras Theorem

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Our triangle has sides A, B, hypotenuse C

Create two squares, the big one with side (A+B) and the small one with side C fitted within big one at 450 angles, making 4 equal triangles on the extra area of the big square with side A, B, and hypotenuse C.

We start by creating squares to help prove that Pythagoras Theorem is true.

#1 calculate the area of the big square by squaring (A+B)

We get the area of the big square as A2+ 2AB + B2

This is the area of the bigger square using the sum of A and B, the area of the square is the same, no matter how we get it.

In #2, we compute the area of the four triangles and the small square.

We apply the formula for calculating the area of a triangle. The are of 1 triangle is ½*AB, since we have 4 triangles, the area will be 2AB. The area of the small square is C2

Next, finding the are of the big square again, add the area of the 4 triangles and that of the small triangle

In #3, we are adding area of small square and that of four triangles.

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