30-60-90 Triangle: The Special Right Triangle Every Student Must Know
Calculate all sides, area, and perimeter of a 30-60-90 special right triangle. Learn the 1 : √3 : 2 ratio and solve any problem with our free calculator.
The 30-60-90 triangle is one of the two special right triangles (alongside the 45-45-90) that form the foundation of trigonometry. Its sides are always in the ratio 1 : √3 : 2, a relationship that emerges naturally from bisecting an equilateral triangle. This triangle appears everywhere — from the construction of hexagonal nut heads and bolt patterns to the geometry of roof trusses and the unit circle. Mastering the 30-60-90 triangle means mastering the fundamental trigonometric values for 30° and 60° angles.
Key Takeaway
In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). If the short leg is a, then the long leg is a√3 and the hypotenuse is 2a. The area is (a²√3)/2.
1. Deriving the 30-60-90 Ratio
The 30-60-90 triangle arises from bisecting an equilateral triangle. An equilateral triangle with side length 2a has all angles equal to 60°. When you draw an altitude from one vertex to the opposite side, the altitude bisects the base and the vertex angle, creating two congruent right triangles with angles 30°, 60°, and 90°.
The altitude has length a√3 (by the Pythagorean theorem: h² + a² = (2a)², so h = a√3). This gives the side ratio: short leg = a, long leg = a√3, hypotenuse = 2a.
Long leg (opposite 60°): a√3
Hypotenuse (opposite 90°): 2a
2. Solving for Any Known Side
Given any one side of a 30-60-90 triangle, you can find the other two using the ratio relationships:
- From short leg a: Long leg = a√3, Hypotenuse = 2a
- From long leg a√3: Short leg = a√3/√3 = a, Hypotenuse = 2a√3/√3 = 2a
- From hypotenuse 2a: Short leg = a, Long leg = a√3
Worked Example
A 30-60-90 triangle has a short leg of length 7. Find all sides, area, and perimeter.
- Short leg = 7
- Long leg = 7√3 ≈ 12.124
- Hypotenuse = 14
- Area = (1/2)(7)(7√3) = 49√3/2 ≈ 42.435
- Perimeter = 7 + 7√3 + 14 = 21 + 7√3 ≈ 33.124
3. Trigonometric Values
The 30-60-90 triangle gives us the exact trigonometric values for 30° and 60°:
| Function | 30° | 60° |
|---|---|---|
| sin | 1/2 | √3/2 |
| cos | √3/2 | 1/2 |
| tan | 1/√3 = √3/3 | √3 |
4. Applications
The 30-60-90 triangle appears in hexagonal geometry (bolt heads, honeycombs), roof pitch calculations, ramp design, clock face geometry, and vector decomposition in physics. Its ratio is fundamental to the construction of equilateral triangles and regular hexagons.
5. Frequently Asked Questions
Can a 30-60-90 triangle be isosceles?
No. The sides are always in ratio 1 : √3 : 2, so all three sides have different lengths. The 30-60-90 triangle is always scalene.
Using Our Calculator
Select which side you know (short leg, long leg, or hypotenuse), enter its value, and the calculator instantly finds all remaining sides, area, and perimeter using the 1 : √3 : 2 ratio.