30-60-90 Triangle Calculator

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

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30-60-90 Triangle Calculator

Enter one side of a 30-60-90 triangle to find all other measurements using the ratio a : a√3 : 2a

a : a\sqrt{3} : 2a = 1 : \sqrt{3} : 2

30-60-90 triangle side ratio

Short Leg (a) - opposite 30°

Shortest side of the triangle

Long Leg (b) - opposite 60°

Length = a√3

Hypotenuse (c) - opposite 90°

Length = 2a

Units

Display Mode

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

a = 1

b = ?

c = ?

Unit 30-60-90 triangle

Example

a = ?

b = 6

c = ?

Given long leg

Example

a = ?

b = ?

c = 10

Given hypotenuse

Example

a = 3

b = ?

c = ?

Small triangle example

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 : 2 (short leg : long leg : hypotenuse)
Short leg (a): Opposite the 30° angle
Long leg (b): Opposite the 60° angle, equals a√3
Hypotenuse (c): Opposite the 90° angle, equals 2a

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 = a\sqrt{3}

c = 2a

Given long leg (b):

a = \frac{b}{\sqrt{3}}

c = \frac{2b}{\sqrt{3}}

Given hypotenuse (c):

a = \frac{c}{2}

b = \frac{c\sqrt{3}}{2}

Memory Tip: The hypotenuse is always twice the short leg, and the long leg is the short leg times √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
• Ratio: 1 : √3 : 2 ✓

Trigonometry Proof

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

\sin(30°) = \frac{1}{2} = \frac{a}{c}

\cos(30°) = \frac{\sqrt{3}}{2} = \frac{b}{c}

For unit hypotenuse: a = 1/2, b = √3/2
Scaling by 2: a = 1, b = √3, c = 2 ✓

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