Pythagorean Theorem Calculator 🌎

The Theorem

a² + b² = c²

Where:

Finding the Hypotenuse

c = sqrt(a² + b²)

Finding a Leg

a = sqrt(c² - b²)
b = sqrt(c² - a²)

Famous Pythagorean Triples

Sets of three whole numbers that perfectly satisfy the theorem:

The 3-4-5 triple is the most useful in construction. If a triangle has sides of exactly 3, 4, and 5 units, it's a right triangle -- this lets you verify right angles with just a tape measure.

Generating Triples (Euclid's Formula)

a = m² - n²
b = 2mn
c = m² + n²

With m > n > 0 as integers. Example: m=2, n=1 gives (3, 4, 5).

Calculating Angles

Once you have all three sides, the acute angles are:

alpha = arctan(a / b)
beta = arctan(b / a)
alpha + beta = 90° (all three angles sum to 180°, one is 90°)

Real-World Applications

Construction

Checking right angles: measure 3 ft and 4 ft along two walls -- the diagonal must be exactly 5 ft. If it's 4.95 or 5.05, the walls aren't square. Used by carpenters and builders for 2,500 years.

Ladder length: ladder against a wall, height 4 m, base 1 m from wall. Length = sqrt(1² + 4²) = sqrt(17) = 4.12 m.

Screen Sizes

A 55-inch TV means 55" diagonal. For a 16:9 aspect ratio:

Distance Between Points

Straight-line distance between two points (x1, y1) and (x2, y2):

d = sqrt((x2 - x1)² + (y2 - y1)²)

This is the Pythagorean theorem applied to Cartesian coordinates. In 3D:

d = sqrt((x2-x1)² + (y2-y1)² + (z2-z1)²)

The Converse Theorem

If a² + b² = c² in a triangle (where c is the longest side), then the triangle IS a right triangle. This lets you verify right angles by measuring sides only.

Generalization: Law of Cosines

For non-right triangles:

c² = a² + b² - 2ab x cos(C)

If C = 90°, then cos(C) = 0 and it reduces to the Pythagorean theorem.

Common Mistakes

1. Applying it to non-right triangles: only works with a 90-degree angle. For others, use the Law of Cosines.
2. Confusing legs with hypotenuse: the hypotenuse is ALWAYS the longest side, opposite the right angle.
3. Forgetting the square root: c² = a² + b², so c = sqrt(a² + b²) (not a + b).
4. Adding before squaring: (a + b)² != a² + b². Square each leg separately, then add.
5. Mixing units: if one leg is in meters and another in centimeters, convert first.