📚 Before We Start
Most geometry books just hand you the 30-60-90 and 45-45-90 ratios and say “memorize these.” But where do they actually come from? It turns out both ratios can be derived completely from scratch using a single idea you already know: the Pythagorean Theorem.
- Pythagorean Theorem: In any right triangle, a2 + b2 = c2 (c = hypotenuse).
- Angles in any triangle sum to 180°.
- An equilateral triangle has all sides equal and all angles equal to 60°.
- A square has all sides equal and all angles 90°.
Both proofs use the same strategy: start with a familiar, highly symmetric shape (an equilateral triangle or a square), then cut it along a line of symmetry. The symmetry tells you everything about the angles for free, and the Pythagorean Theorem fills in the side lengths.
Part 1 The 30-60-90 Triangle
Sides are always in the ratio x : x√3 : 2x
Part 2 The 45-45-90 Triangle
Sides are always in the ratio x : x : x√2
📋 Summary
30-60-90
| Angle | Opposite Side | Relative Size |
|---|---|---|
| 30° | x | shortest |
| 60° | x√3 ≈ 1.73x | middle |
| 90° | 2x | longest (hyp) |
45-45-90
| Angle | Opposite Side | Relative Size |
|---|---|---|
| 45° | x | shorter (leg) |
| 45° | x | shorter (leg) |
| 90° | x√2 ≈ 1.41x | longest (hyp) |
Both proofs follow the same three steps:
1. Start with a perfectly symmetric shape.
2. Cut it along a line of symmetry — symmetry tells you the angles for free.
3. Use the Pythagorean Theorem to find the missing side.
No sine. No cosine. No tangent. Just symmetry and one familiar formula.
✏️ Try It Yourself
Find the missing sides. Use the ratio, then click to check.