02 // Visual Math & Finance SolversArea(a²) + Area(b²) = Area(c²) ⇒ c = √(a² + b²)
Visual Pythagorean Theorem Geometric Area Proof ($a^2 + b^2 = c^2$)
The Pythagorean Theorem states that in any Euclidean right triangle, the combined surface area of the squares constructed on legs $a$ and $b$ equals the exact surface area of the square constructed on hypotenuse $c$: $a^2 + b^2 = c^2$.
Area(a²) + Area(b²) = Area(c²) ⇒ c = √(a² + b²)Inputs & Parameters (2)
u
1u±0.525u
u
1u±0.525u
Hypotenuse c = √(a²+b²)
10
c² = 64 + 36 = 100
Acute Angles (α, β)
36.87° / 53.13°
Grade = 75% slope
Area & Perimeter
24 u²
Perimeter P = 24 u
Altitude h_c & Inradius r
h_c = 4.8
Inradius r = 2
20-Row Comparison Table
Compare across:
| # | Leg a Length (u) | Hypotenuse (c) | Angle α | Area (½ab) |
|---|---|---|---|---|
| 01 | 1 u | 6.083 u | 80.54° | 3 u² |
| 02 | 2.5 u | 6.5 u | 67.38° | 7.5 u² |
| 03 | 3.5 u | 6.946 u | 59.74° | 10.5 u² |
| 04 | 5 u | 7.81 u | 50.19° | 15 u² |
| 05 | 6 u | 8.485 u | 45° | 18 u² |
| 06 | 7.5 u | 9.605 u | 38.66° | 22.5 u² |
| 07 | 8.5 u | 10.404 u | 35.22° | 25.5 u² |
| 08 | 10 u | 11.662 u | 30.96° | 30 u² |
| 09 | 11 u | 12.53 u | 28.61° | 33 u² |
| 10 | 12.5 u | 13.865 u | 25.64° | 37.5 u² |
| 11 | 13.5 u | 14.773 u | 23.96° | 40.5 u² |
| 12 | 15 u | 16.155 u | 21.8° | 45 u² |
| 13 | 16 u | 17.088 u | 20.56° | 48 u² |
| 14 | 17.5 u | 18.5 u | 18.92° | 52.5 u² |
| 15 | 18.5 u | 19.449 u | 17.97° | 55.5 u² |
| 16 | 20 u | 20.881 u | 16.7° | 60 u² |
| 17 | 21 u | 21.84 u | 15.95° | 63 u² |
| 18 | 22.5 u | 23.286 u | 14.93° | 67.5 u² |
| 19 | 23.5 u | 24.254 u | 14.32° | 70.5 u² |
| 20 | 25 u | 25.71 u | 13.5° | 75 u² |
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Key Analytical Takeaways
- 01When $a = 8$ and $b = 6$, the leg squares have areas $64$ and $36$, summing to $100$—matching the $10 \times 10$ square on hypotenuse $c = 10$.
- 02The altitude dropped from the right angle splits the hypotenuse square $c^2$ into two rectangles of areas $a^2$ and $b^2$ (Euclid's Windmill proof).
- 03If $a = b$, the hypotenuse simplifies to the diagonal of a square: $c = a\sqrt{2} \approx 1.4142a$.
Frequently Asked Questions
Does the Pythagorean theorem work on non-right triangles?+
For non-right triangles, it generalizes into the Law of Cosines: c² = a² + b² − 2ab·cos(C), which reduces to a² + b² = c² when C = 90° because cos(90°) = 0.
What are primitive Pythagorean triples?+
Sets of three coprime positive integers (a, b, c) satisfying a² + b² = c², generated by (m² − n², 2mn, m² + n²) for integers m > n > 0.
What is a primitive Pythagorean triple?+
A primitive Pythagorean triple is a set of three coprime positive integers (a, b, c) satisfying a² + b² = c², such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).