Theorems are arguments you trust
A theorem is a statement that's been proven once and for all. The Pythagorean theorem is famous because it shows up everywhere — distances, navigation, computer graphics — and you only need a right triangle to use it.
Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c: a² + b² = c². The square on the hypotenuse equals the sum of the squares on the legs.
Famous triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Distance and midpoint
Distance: d = √((x₂−x₁)² + (y₂−y₁)²) — Pythagoras in disguise.
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).
Triangle congruence
Two triangles are congruent if matching parts force every other part. Postulates: SSS, SAS, ASA, AAS, HL.
Special right triangles
45-45-90: sides in ratio 1 : 1 : √2.
30-60-90: sides in ratio 1 : √3 : 2.
Quick reference
Hands-on tools
Drag the legs of a right triangle, slide a sphere's radius, or check distances on a grid.
Pythagorean theorem visualizer
The squares on the legs add up to the square on the hypotenuse.
3D volume calculator
Pick a solid and slide the dimensions.
Coordinate distance walker
Pick two points on the grid. Distance shows the right triangle behind it.
Triangle congruence postulate matcher
For each scenario, pick which postulate proves the triangles congruent.
Quiz
Flashcards
Tap a card to flip. ← / → keys to navigate.
Daily challenge
A new problem every day. Same problem for everyone, worldwide. Build a streak — one shot per day.
For teachers
Print-ready worksheet, answer key, teaching tips and standards alignment.
Teaching tips
Standards alignment
Reference
Formula sheet
Photo gallery
Images sourced from Wikipedia.