Pythagorean Theorem

Pythagorean Theorem Diagram
Pythagorean Theorem

The Pythagorean Theorem is probably the most well known geometric theorem. It says that in a right triangle, the sum of the squares of the sides, called legs, will equal the square of the hypotenuse [the slant side.] A² + B² = C².

Pythagorean Theorem Diagram Example: if the legs of a right triangle are 3 and 4, then 3x3=9, and 4x4=16, and 9+16=25, and so the hypotenuse is 5.

Archeologists use the Pythagorean Theorem in field excavations. When they start a dig, they place a rectangular grid over the site surface. To lay out an accurate grid system, archeologists use the Theorem, X² + Y² = D². After deciding how long the baselines [X- and Y- axis] should be, the proper length of the diagonal is calculated using the Pythagorean theorem to make sure the quadrant is a rectangle. Corner stakes will then be placed to mark the accurate location of the site.

Pythagoras Who?

Pythagoras was an influential Greek mathematician and philosopher, best known for the theory to which he gave his name.


Very little is known about Pythagoras's life. He is thought to have been born on the Greek island of Samos approximately 569 BC.

He travelled widely in his youth, visiting Egypt and Persia. He settled in the city of Crotone in southern Italy. There he began teaching and soon had a clutch of students who lived a structured life of study and exercise, inspired by a philosophy based around mathematics. This circle came to be known as the Pythagoreans.

Pythagoreans believed that everything could be reduced to numbers: the whole universe had been built using mathematics. The truth behind the everyday reality we experience lay in numbers.

Find the Third Side

Right Triangle
Given A=6 and B=4, find C.
A² + B² = C²
6² + 4² = C²
36 + 16 = C²
C² = 52
C = √ 52 = 7.2

Given A=8 and C=12, find B.
A² + B² = C²
8² + B² = 12²
B² = 12² - 8²
B² = 144 - 64
B = √ 80 = 8.9

Visual Proof

Visual Proof
Study the grid. Find the right triangle outlined in red. See the squares built on the sides - yellow, blue, and green. Count the number of triangles in each square - yellow (4), blue (4) and green (8). Therefore, the green square equals the sum of the yellow square + blue square.

Self Quiz

1. What is the name of a right triangle's slant side?

2. Is this triangle a right triangle? Why or why not?
Right Triangle

3. Find the missing side.
Right Triangle


1. hypotenuse
2. yes, 25 + 144 = 169
3. 41