Geometry · Right Triangles
Pythagorean
Theorem Calculator
Enter any 2 values of a right triangle — solve for the missing side, all angles, trig ratios, area, and check for Pythagorean triples. Includes SVG diagram and visual proofs.
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Hypotenuse c
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Area
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Angle A
Solve: a² + b² = c²
Enter any 2 values. Leave the missing side blank — it will be calculated. c is always the hypotenuse (longest side).
Famous Triples (click to load)
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Right Triangle Diagram
Hypotenuse c
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Area
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Angle A
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Angle B
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📐 Complete Right Triangle Properties
📊 Trigonometric Ratios
📋 Step-by-Step Solution
🔢 Famous Pythagorean Triples
Any multiple of a triple is also a triple: 3-4-5 × 2 = 6-8-10, × 3 = 9-12-15, etc. There are infinitely many primitive triples (with no common factor). Generated by: a=m²−n², b=2mn, c=m²+n² for integers m>n>0.
🎨 Three Visual Proofs of the Pythagorean Theorem
1. Square on Hypotenuse
The square built on the hypotenuse (c²) has exactly the same area as the two squares built on the legs (a² + b²). Rearrange 4 copies of the right triangle to prove it — shown in both arrangements.
2. Similar Triangles
Drop the altitude from the right angle to the hypotenuse. This creates 3 similar triangles (original + two smaller). The proportions of similar triangles immediately give a² + b² = c² — one of the most elegant proofs.
3. Garfield's Proof (1876)
US President James Garfield discovered this proof in 1876. Arrange two right triangles and a third isosceles triangle into a trapezoid. Area of trapezoid = sum of three triangles → ½(a+b)² = ½ab + ½ab + ½c² → a² + b² = c².
The Pythagorean Theorem: a² + b² = c²
The Pythagorean theorem is one of the most proven theorems in mathematics — with over 370 known proofs. It states that in any right triangle, the area of the square on the hypotenuse equals the sum of areas of squares on the two legs.
Right Triangle Formulas
Given legs a, b: c = √(a² + b²)
Given a and c: b = √(c² − a²)
Area: A = ½ × a × b
Perimeter: P = a + b + c
Angle A (opp a): A = arctan(a/b) = arcsin(a/c)
Angle B (opp b): B = arctan(b/a) = arcsin(b/c)
A + B = 90° (all acute angles sum to 90° in right Δ)
Special right triangles:
45-45-90: legs a, a → hypotenuse = a√2
30-60-90: short leg a → long leg = a√3, hypotenuse = 2a
How are Pythagorean triples generated?
All primitive Pythagorean triples (no common factor) are generated by the formula: a = m²−n², b = 2mn, c = m²+n², where m > n > 0, gcd(m,n)=1, and m and n have opposite parity (one even, one odd). For m=2, n=1: a=3, b=4, c=5. For m=3, n=2: a=5, b=12, c=13. For m=4, n=1: a=15, b=8, c=17. Any multiple of a primitive triple is also a triple.
What are the special right triangles?
Two special right triangles appear constantly in math and physics. The 45-45-90 triangle has legs in ratio 1:1 and hypotenuse = leg×√2. The 30-60-90 triangle has sides in ratio 1:√3:2. They come from cutting a square diagonally (45-45-90) or cutting an equilateral triangle in half (30-60-90). Their trig values are exact: sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2.
Does the Pythagorean theorem work in 3D?
Yes — the 3D distance formula is a direct extension: for a rectangular box with dimensions l, w, h, the space diagonal d = √(l²+w²+h²). This is applying the theorem twice: first find the floor diagonal √(l²+w²), then use it as a leg with h to find d. In n dimensions: d = √(x₁²+x₂²+...+xₙ²). This is the Euclidean norm, fundamental to geometry in any number of dimensions.
Who proved the Pythagorean theorem first?
Despite being named after Pythagoras (~570–495 BCE), the relationship was known earlier. The Babylonian tablet Plimpton 322 (c. 1800 BCE) lists Pythagorean triples, and ancient Indian (Sulbasutras, c. 800 BCE) and Chinese (Zhoubi Suanjing) texts describe the relationship. Pythagoras or his school likely provided the first formal proof. Today, over 370 proofs exist — by Euclid (Elements Book I), Garfield, Leonardo da Vinci, and thousands of high school students worldwide.