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30-60-90 Triangle Calculator

30-60-90 Triangle Calculator

Solve 30-60-90 special right triangles using the ratio 1:3:21 : \sqrt{3} : 2. Find missing sides, area, and perimeter with step-by-step solutions and visual triangle diagrams.

100% FreeStep-by-step SolutionsVisual Triangle
30-60-90 Triangle Calculator
Enter one side of a 30-60-90 triangle to find all other measurements using the ratio a:a3:2aa : a\sqrt{3} : 2a
Enter to calculate, Esc to clear
a:a3:2a=1:3:2a : a\sqrt{3} : 2a = 1 : \sqrt{3} : 2

30-60-90 triangle side ratio

Shortest side of the triangle

Length = a3a\sqrt{3}

Length = 2a2a

Instructions: Enter only one side length. The calculator will compute the other two sides using the 30-60-90 triangle ratio.
Try These Examples
Click on any example to automatically fill the calculator
Example

Unit 30-60-90 triangle

a: 1
b: ?
c: ?
Example

Given long leg

a: ?
b: 6
c: ?
Example

Given hypotenuse

a: ?
b: ?
c: 10
Example

Small triangle example

a: 3
b: ?
c: ?
What Is a 30-60-90 Triangle?

A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. It has a fixed side ratio that makes calculations predictable and efficient.

Key Properties:

  • Angles: 30°, 60°, 90° (sum = 180°)
  • Side Ratio: 1:3:21 : \sqrt{3} : 2 (short leg : long leg : hypotenuse)
  • Short leg (a): Opposite the 30° angle
  • Long leg (b): Opposite the 60° angle, equals a3a\sqrt{3}
  • Hypotenuse (c): Opposite the 90° angle, equals 2a2a

Origin: This triangle appears when you split an equilateral triangle in half along its altitude.

How to Use the 30-60-90 Triangle Ratio

Calculation Method:

Given short leg (a):
b=a3b = a\sqrt{3}
c=2ac = 2a
Given long leg (b):
a=b3a = \frac{b}{\sqrt{3}}
c=2b3c = \frac{2b}{\sqrt{3}}
Given hypotenuse (c):
a=c2a = \frac{c}{2}
b=c32b = \frac{c\sqrt{3}}{2}

Memory Tip: The hypotenuse is always twice the short leg, and the long leg is the short leg times sqrt3\\sqrt{3}.

Real-World Applications

Architecture & Construction

  • Roof pitch calculations (30° slopes)
  • Stair design with specific angles
  • Triangular structural supports
  • Window and door frame angles
  • Hexagonal tile patterns

Engineering & Design

  • Mechanical component design
  • Solar panel optimal angles
  • Bridge truss calculations
  • Gear tooth geometry
  • Optical system angles

Mathematics & Education

  • Trigonometry problem solving
  • Geometry proofs and theorems
  • Competition math shortcuts
  • Physics vector calculations
  • Navigation and surveying
Proofs of 30-60-90 Triangle Ratios

Equilateral Triangle Method

Start with an equilateral triangle with side length 2. Draw an altitude from one vertex to the opposite side, creating two congruent 30-60-90 triangles.

• Original triangle side = 2
• Altitude creates two sides of length 1
• Height = 22−12=3\sqrt{2^2 - 1^2} = \sqrt{3}
• Ratio: 1:3:21 : \sqrt{3} : 2 ✓

Trigonometry Proof

Using basic trigonometric ratios with a unit hypotenuse (c = 1):

sin⁡(30°)=12=ac\sin(30°) = \frac{1}{2} = \frac{a}{c}
cos⁡(30°)=32=bc\cos(30°) = \frac{\sqrt{3}}{2} = \frac{b}{c}
For unit hypotenuse: a=12,b=32a = \frac{1}{2}, b = \frac{\sqrt{3}}{2}
Scaling by 2: a=1,b=3,c=2a = 1, b = \sqrt{3}, c = 2 ✓
Frequently Asked Questions
A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. Its sides follow the ratio 1:√3:2, making calculations predictable without complex trigonometry.
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30-60-90 Triangle Calculator - Find Sides, Area & Perimeter | MathIsimple