Special Right Triangles: 30-60-90
A 30-60-90 triangle has sides in the ratio . It comes from cutting an equilateral triangle in half.
Introduction
Draw an altitude from any vertex of an equilateral triangle to the opposite side. You bisect the base and the apex angle, creating two 30-60-90 triangles. This is the most common “special” right triangle and appears constantly in trigonometry, the unit circle, and standardized tests.
Past Knowledge
45-45-90 (8.2.1). Pythagorean Theorem. Equilateral triangles.
Today's Goal
Find all sides of a 30-60-90 triangle given any one side.
Future Success
SOH CAH TOA (8.2.3), trig ratios (8.2.4–8.2.5), unit circle.
Key Concepts
The 30-60-90 Ratio
- Short leg () is opposite the 30° angle
- Long leg () is opposite the 60° angle
- Hypotenuse () is opposite the 90° angle — always twice the short leg
Theorem & Proof
Derivation: 30-60-90 Side Ratio from an Equilateral Triangle
Given: Equilateral triangle with side . Drop altitude from one vertex to the opposite side.
Derive: The side lengths of the resulting 30-60-90 triangle.
| # | Statement | Reason |
|---|---|---|
| 1 | Equilateral triangle has all angles = 60° and all sides = | Definition of equilateral triangle |
| 2 | The altitude bisects the base and the apex angle | In an equilateral triangle, the altitude is also a median and angle bisector |
| 3 | Short leg = , hypotenuse = , angles = 30°, 60°, 90° | Bisected base and angle (steps 1–2) |
| 4 | → | Pythagorean Theorem on the half-triangle |
| 5 | Positive square root |
∎ Sides: . The hypotenuse is always double the short leg.
Worked Examples
Short Leg Given
Short leg = 5. Find the other sides.
Long leg = , Hypotenuse =
Long leg = , Hyp = 10
Hypotenuse Given
Hypotenuse = 14. Find both legs.
Short leg =
Long leg =
Short = 7, Long =
Long Leg Given
Long leg = . Find the other sides.
Long leg = →
Short leg = 6, Hypotenuse =
Short = 6, Hyp = 12
Common Pitfalls
Mixing Up Which Leg Gets √3
The goes on the long leg (opposite 60°), not the short one. The short leg is just , and the hypotenuse is .
Confusing 45-45-90 and 30-60-90
45-45-90 uses for the hypotenuse. 30-60-90 uses for the long leg and 2 for the hypotenuse. Keep them separate!
Real-Life Applications
Roofs — Standard Pitch
A “12 in 12” roof pitch creates 45° angles, but “7 in 12” pitches closely approximate 30° — common in residential construction. Roofers use the 30-60-90 ratios to calculate rafter lengths.
Hexagons in Nature
Honeycomb cells are regular hexagons, which are made of six equilateral triangles. Every calculation involving hexagon height, apothem, and area uses the 30-60-90 ratio.
Practice Quiz
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