30-60-90 Triangle Calculator
Calculate sides, area, and perimeter of special 30-60-90 right triangles
30-60-90 Triangle Calculator
30-60-90 Triangle Diagram
Example Calculation
Construction Triangle Example
Given: Long leg (opposite 60°) = 11 inches
Type: 30-60-90 triangle
This is a special right triangle commonly used in construction and drafting.
Solution
Short leg: a = b/√3 = 11/√3 ≈ 6.35 inches
Long leg: b = 11 inches (given)
Hypotenuse: c = 2b/√3 = 22/√3 ≈ 12.70 inches
Area: A = (a²√3)/2 ≈ 34.9 square inches
Perimeter: P = a(3 + √3) ≈ 30.05 inches
30-60-90 Triangle Properties
Special Right Triangle
Angles: 30°, 60°, 90°
Half of an equilateral triangle
Fixed Ratios
Sides: 1 : √3 : 2
Consistent proportions
Practical Applications
Architecture, engineering
Roof pitches, set squares
Formula Reference
Side Ratios
a : a√3 : 2a
From Short Leg (a)
b = a√3, c = 2a
From Long Leg (b)
a = b/√3, c = 2b/√3
From Hypotenuse (c)
a = c/2, b = c√3/2
Area
A = (a²√3)/2
Perimeter
P = a(3 + √3)
Understanding 30-60-90 Triangles
What Makes This Triangle Special?
A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. It's actually half of an equilateral triangle, which gives it unique and predictable side ratios that make calculations simple and exact.
Key Properties
- •Side ratio is always 1 : √3 : 2
- •Only right triangle with angles in arithmetic progression
- •Derived from cutting an equilateral triangle in half
- •Exact values involve √3
Real-World Applications
- •Architecture and roof design
- •Drafting and technical drawing
- •Engineering calculations
- •Set squares and measuring tools
- •Trigonometry problems
Memory Tip: The side ratios 1 : √3 : 2 correspond to the sides opposite angles 30° : 60° : 90° respectively. The shortest side is always opposite the smallest angle.
Mathematical Derivation
From Equilateral Triangle
Start with equilateral triangle with side length 2a
Height = side × √3/2 = 2a × √3/2 = a√3
When cut in half: short leg = a, long leg = a√3, hypotenuse = 2a
Using Trigonometry
sin(30°) = 1/2, so a/c = 1/2, therefore c = 2a
sin(60°) = √3/2, so b/c = √3/2, therefore b = a√3
This confirms the ratio a : a√3 : 2a