The Pythagorean Theorem states that the squared lengths of the two legs on a right triangle added to one another equal the length of the hypotenuse squared. In other words, if a square were drawn onto each side of a right triangle, the sum of the areas from the two smaller squares would equal the area of the largest square (Posamentier). The formula which represents this idea is: a^{2} + b^{2} = c^{2}. There have been hundreds of proofs for this foundational theorem throughout the years. This page displays just a few of those proofs, demonstrating just how diverse the theorem is.

An interesting proof of the theorem is believed to have originated in China, though Pythagoras may have discovered it as well. It consists of tangram-like images that can be understood by viewing the visual representation and don't require mathematical derivation or calculation. A description of this visual proof of the theorem reads, "The shorter leg multiplied by itself is the red square, and the longer leg multiplied by itself is the blue square. Let them be moved about so as to patch each other, each 2 according to its type. Because the differences are completed, there is no instability. They form together the area of the square on the hypotenuse; extracting the square root gives the hypotenuse." (Overduin)

Another proof from Asia gives additional insight on the theorem. Using four equivalent right triangles where the two legs are of different length, positioning them will form a square with the hypotenuse as the side lengths. A smaller square remains in the center of the original square formed. The following can be observed from the image created:

Area of c square = Area of triangles + Area of small square

Rearranging the triangles and small square will reveal the relationship of the Pythagorean Theorem, as it can now be observed:

Area of c square= Area of a square + Area of b square

It can be concluded that c^{2} = a^{2} + b^{2} (Wijayanti).

From Euclid's geometric proof there was a somewhat algebraic version developed. This proof was recorded in a textbook which was eventually translated into English by a man named Charles Davis. His proof became very popular throughout the world, it was written as follows. Beginning with △ ABC, where ∠ ACB is the right angle, a segment CD is created perpendicular to the hypotenuse, AB. This ensures right angles in ∠ ADC and ∠ BDC. It is also noted that ∠ DAC = ∠ DCB and ∠ ACD = ∠ DBC. The triangles △ ADC and △ CDB are similar to each other because their angles are congruent, and △ ABC is similar to them as well. Thus, AC/AB = AD/AC and BC/AB = BD/BC. Therefore,

AC^{2} = AB × AD and BC^{2} = AB × BD.

By adding the previous relations, we get

AC^{2} + BC^{2} = AB (AD + BD) = AB × AB = AB^{2} (Agarwal).

The proof which President Garfield developed relies on the facts that the sum of all angles of a triangle is 180 degrees, and the area of a trapezoid of base lengths b_{1} and b_{2} and height h is indeed A = ½ (b_{1}+b_{2})h. His proof is based on an image of a right triangle with side lengths a and b with hypotenuse length c, where the triangle is copied and positioned as shown in the picture below. A segment is also constructed to connect the triangles, forming a trapezoid.

Starting with the construction of the trapezoid, his proof is related as the following: "The parallel sides of the trapezoid (which are the left and right sides in the figure) have lengths a and b. The height of the trapezoid (which is the distance from left to right in the figure) is a + b. Thus the area of the trapezoid is

A = ½(a + b)(a + b)= ½ (a + b)^{2} .

However, the area of the trapezoid is also the sum of the areas of the three triangles that make up the trapezoid. Note that the middle triangle is also a right triangle because A + B = 90^{o}. The area of the trapezoid is thus

A = ½ a b + ½ a b + ½ cc = ab + ½ c^{2}.

We thus conclude that

½ (a + b)^{2} = ab+ ½ c^{2}.

Multiplying both sides of this equation by 2 gives us

(a+b)^{2} = 2ab + c^{2}.

Expanding the left hand side of the above equation then gives

a^{2} + 2ab + b^{2} = 2ab + c^{2}

from which we arrive at the conclusion that

a^{2} + b^{2} = c^{2}" (Ellermeyer).

The above proofs are evidence that the Pythagorean Theorem is foundational to geometry. Mathematicians throughout the years took time proving it, which implies it is just as important as it was centuries ago when it was discovered. Many have continued the search for new ways to prove it. Each proof gives its own addition to understanding the theorem and when all are studied together, the Pythagorean Theorem shines in full light. The following applet is designed to guide you in an exploration of the Pythagorean Theorem. Follow the provided instructions, answer the listed questions, and make connections between the proofs as you further explore this exciting theorem.

Move points A and B to look at different right triangles.

1. Click the squares check box. How do the areas of the blue squares relate to the pink square?

2. Use the side lengths of the triangle in the formula given. Check your response with the side lengths check box.

3. Use the area of the squares in place of the side lengths squared. Check your response with the areas check box.

4. What happens to the length of c when a or b decreases? increases?

5. What can you conclude about the side lengths of right triangles from this activity?