Pythagorean Theorem Visualized
The Pythagorean theorem is one of the most famous results in all of mathematics: for any right triangle with legs a and b and hypotenuse c,
a² + b² = c²
Let’s make this concrete by playing with actual numbers.
Pick Your Leg Lengths
Use the sliders to set the two legs of a right triangle. The hypotenuse is calculated automatically.
Classic triple: Set a = 3 and b = 4. You should get c = 5 — the most famous Pythagorean triple. Try a = 5, b = 12 for another one (c = 13).
Graphing the Right Triangle
Below, the graph plots a right triangle with one leg along the x-axis (length a) and one leg along the y-axis (length b). The hypotenuse connects (a, 0) to (0, b).
The hypotenuse is the line from (0, b) to (a, 0). Its length is exactly sqrt(a² + b²) — that is the Pythagorean theorem in action.
The Area Relationship
The theorem is really about areas. If you draw a square on each side of the triangle:
- The square on leg a has area a²
- The square on leg b has area b²
- The square on the hypotenuse has area c² = a² + b²
The two smaller squares together have the exact same area as the big square.
The red line (c²) always equals the sum of the blue (a²) and cyan (b²) lines. Slide the legs around and watch the areas change while the relationship holds.
Beyond right triangles: If a² + b² > c², the triangle is acute (all angles < 90°). If a² + b² < c², it is obtuse (one angle > 90°). The Pythagorean theorem is the boundary case — the right angle case.
How c Changes as You Adjust the Legs
This graph shows c = sqrt(x² + b²) as a function of x, with b set by your slider. It is not a straight line — it is a curve that grows more slowly as a gets larger.
Challenge: A ladder leans against a wall. The foot of the ladder is 6 feet from the wall and the ladder is 10 feet long. How high up the wall does the ladder reach? Use the sliders to verify your answer. (Hint: the ladder is the hypotenuse.)