What is the ratio of sides in a 30-60-90 triangle?

The side ratio is 1:√3:2 or equivalently a:a√3:2a. If the short leg (opposite 30°) is 'a', then the long leg (opposite 60°) is a√3 ≈ 1.732a, and the hypotenuse (opposite 90°) is 2a.

How do I find the sides of a 30-60-90 triangle?

Given short leg (a): long leg = a√3, hypotenuse = 2a. Given long leg (b): short leg = b/√3, hypotenuse = 2b/√3. Given hypotenuse (c): short leg = c/2, long leg = c√3/2. Just enter one known side in our calculator.

How is a 30-60-90 triangle formed?

A 30-60-90 triangle is formed by cutting an equilateral triangle in half with an altitude. This creates two congruent 30-60-90 triangles. The altitude becomes the long leg, half the original side becomes the short leg, and the original side becomes the hypotenuse.

What is the area formula for a 30-60-90 triangle?

Area = (1/2) × short leg × long leg = (1/2) × a × a√3 = (a²√3)/2. If you know the hypotenuse c, Area = (c²√3)/8. Our calculator computes this automatically when you enter any side.

Where are 30-60-90 triangles used in real life?

30-60-90 triangles appear in architecture (roof pitches, hexagonal patterns), engineering (gear teeth, truss designs), navigation, solar panel angles, and standardized test problems (SAT, ACT, GRE). They're essential in trigonometry and physics calculations.

30-60-90 Triangle Calculator - Find Sides, Area & Perimeter

30-60-90 Triangle Calculator

Solve 30-60-90 special right triangles using the ratio $1 : \sqrt{3} : 2$ . 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 : a\sqrt{3} : 2a

Enter to calculate, Esc to clear

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 = $a\sqrt{3}$

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

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 one of two "special" right triangles you'll see over and over in geometry and trig. Its sides always follow the ratio $1 : \sqrt{3} : 2$ , which means if you know any one side, you can find the other two instantly — no Pythagorean theorem needed.

Key Properties:

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

Origin: Cut any equilateral triangle in half along its altitude and you get two 30-60-90 triangles. That's where the clean ratio comes from.

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 $\sqrt{3}$ .

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

Mastering the 30-60-90 Triangle: From Theory to Practice

Why the 1:√3:2 Ratio Works

The 30-60-90 triangle is born from a perfect geometric construction: splitting an equilateral triangle in half. When you draw an altitude from any vertex of an equilateral triangle to the opposite side, it creates two mirror-image 30-60-90 triangles. This altitude bisects the base perpendicularly, forming the 90° angle.

Starting with an equilateral triangle where all sides equal 2 units: the altitude divides the base into two segments of 1 unit each (the short leg). Using the Pythagorean theorem to find the altitude height: $h^2 + 1^2 = 2^2$ → $h = \sqrt{3}$ (the long leg). The original side becomes the hypotenuse (2 units). This gives us the ratio $1:\sqrt{3}:2$ (approximately 1:1.732:2).

Real-World Example 1: 30° Roof Pitch

Scenario: A residential garage requires a 30° roof slope with a 10-foot horizontal span (run).

• Horizontal run (short leg):

a = 10

• Vertical rise (long leg):

b = 10\sqrt{3} \approx 17.32

• Rafter length (hypotenuse):

c = 2 \times 10 = 20

• Each rafter material needed: 21 ft board (20 ft + overhang)

A 30° pitch is steeper than the common 4:12 pitch (18.4°), providing excellent drainage in rainy climates while remaining walkable for maintenance. Contractors use the 1:√3:2 ratio to quickly calculate that a 10 ft run requires a 17.3 ft rise and 20 ft rafters—no trigonometry calculator needed on-site.

Real-World Example 2: Hexagonal Tile Design

Scenario: Designing a honeycomb tile floor with regular hexagons where each side measures 4 inches.

• Hexagon side length: 4 in (forms hypotenuse c)

• Short leg of internal triangle:

a = 4/2 = 2

• Long leg (triangle height):

b = 2\sqrt{3} \approx 3.46

• Total hexagon height:

2b = 4\sqrt{3} \approx 6.93

Regular hexagons decompose into six equilateral triangles or twelve 30-60-90 triangles. Tile manufacturers use this ratio to design interlocking hexagonal patterns—knowing a 4" side hexagon has a 6.93" vertical span helps calculate how many tiles fit a floor. The long leg $\sqrt{3}/2 \times \text{side}$ determines spacing.

Trigonometric Values Derived from 30-60-90

The 30-60-90 triangle is the geometric foundation for memorizing exact trigonometric values. Using a triangle with sides $1, \sqrt{3}, 2$ :

For 30° angle:

\sin(30°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2} = 0.5

\cos(30°) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \approx 0.866

\tan(30°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} \approx 0.577

For 60° angle:

\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866

\cos(60°) = \frac{1}{2} = 0.5

\tan(60°) = \sqrt{3} \approx 1.732

Notice how sin(30°) = cos(60°) and cos(30°) = sin(60°). This complementary relationship exists because 30° + 60° = 90°, making them complementary angles in the right triangle.

Explore More from Authoritative Resources

Wolfram MathWorld: 30-60-90 Triangle

Rigorous mathematical treatment including proofs, generalizations, and historical context from Wolfram Research.

OpenStax Precalculus: Special Triangles and Trig Identities

Free college-level textbook chapter covering special right triangles with interactive exercises and applications.

Math is Fun: Special Right Triangles

Clear visual explanations and interactive diagrams for both 30-60-90 and 45-45-90 triangles.

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 = $\sqrt{2^2 - 1^2} = \sqrt{3}$
• Ratio: $1 : \sqrt{3} : 2$ ✓

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 = \frac{1}{2}, b = \frac{\sqrt{3}}{2}

Scaling by 2:

a = 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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