# 30 60 90 Triangle: Calculator, Rules, Formula, Properties, Examples

The **30 60 90 triangle** is a unique right triangle with three different angles: 30 degrees, 60 degrees, and 90 degrees. Geometry is based on this unique triangle, which has many remarkable traits and formulas. We will go over the 30 60 90 triangle in great detail here, including how to use a 30 60 90 triangle calculator, the main rules and formula, the characteristics of this unique right triangle, solved examples, and a free worksheet to practice 30 60 90 triangle issues.

This complete introduction to the 30-60-90 triangle contains all you need, regardless of your level of geometry knowledge—teacher, student, or simply inquisitive.

## Part 1. What is 30 60 90 triangle?

A 30 60 90 triangle is a particular right triangle having angles of 30 degrees, 60 degrees, and 90 degrees. Geometry and trigonometry find great relevance in this triangle since it possesses special qualities.

### Properties of 30-60-90 triangle

1 | The angles are 30 degrees, 60 degrees, and 90 degrees. |

2 | The sides are in the ratio of 1:√3:2, where the shortest side is 1, the medium side is √3, and the longest side is 2. |

3 | The hypotenuse is always twice the length of the shortest side. |

4 | The medium side is always √3 times the length of the shortest side. |

### Special Right Triangles 30 60 90

One of the two unique right triangles in geometry; the other is the 45 45 90 triangle. From navigation to building, these unique triangles find application in various fields thanks to their particular qualities and formulas.

## Part 2. 30 60 90 Triangle Calculator

Designed to help you quickly answer problems with this unique right triangle, the 30 60 90 Triangle Calculator is The calculator can find the missing numbers by entering known facts, such the value of one angle or the length of one side.

### 30 60 90 Triangle Calculator

## Part 2. 30 60 90 Triangle Rules

The 30 60 90 Triangle Rules are a set of rules that define the unique characteristics and linkages found within this specific right triangle. A 30-60-90 triangle follows these general guidelines:

1 | The angles are 30 degrees, 60 degrees, and 90 degrees. |

2 | The sides are in the ratio of 1:√3:2, where the shortest side is 1, the medium side is √3, and the longest side is 2. |

3 | The hypotenuse is always twice the length of the shortest side. |

4 | The medium side is always √3 times the length of the shortest side. |

5 | If the shortest side is known, the other two sides can be easily calculated using the ratios. |

For example, if the shortest side of a 30 60 90 triangle is 6 units, then:

- The medium side would be 6√3 ≈ 10.39 units.
- The longest side would be 12 units.

From engineering and construction to navigation and trigonometry, these guidelines make the 30 60 90 triangle a practical instrument in many fields.

## Part 3. 30 60 90 Triangle Formula

The 30 60 90 Triangle Formula describes the mathematical relationships between the sides and angles of this unusual right triangle. These formulas and recommendations for the 30, 60, and 90 triangle provide a comprehensive set of tools for dealing with scenarios involving this peculiar right triangle.

Formula | Description |
---|---|

a = x b = x√3 c = 2x | Where ‘a’ is the shortest side, ‘b’ is the medium side, ‘c’ is the longest side, and ‘x’ is the length of the shortest side. |

sin 30° = 1/2 cos 30° = √3/2 tan 30° = 1/√3 | The trigonometric ratios for the 30-degree angle. |

sin 60° = √3/2 cos 60° = 1/2 tan 60° = √3 | The trigonometric ratios for the 60-degree angle. |

sin 90° = 1 cos 90° = 0 tan 90° = undefined | The trigonometric ratios for the 90-degree angle. |

## Part 4. 30 60 90 triangle side lengths

A 30 60 90 triangle’s side lengths follow a precise ratio that makes this triangle a handy instrument in many different fields.

A 30 60 90 triangle’s side lengths fall in the ratio 1:√3:2, whereby:

- The shortest side has a length of 1 unit.
- The medium side has a length of √3 units.
- The longest side (the hypotenuse) has a length of 2 units.

### What are the ratios of a 30 60 90 Triangle?

The triangle’s angles of 30 degrees, 60 degrees, and 90 degrees produce this ratio directly.

Thirty sixty ninety triangle ratios.

A 30 60 90 triangle has sides with ratios as follows:

- Shortest side (a) : Medium side (b) : Longest side (c) = 1 : √3 : 2
- a = 1 unit
- b = √3 units
- c = 2 units

As long as the angles stay 30 degrees, 60 degrees, and 90 degrees, these ratios are constant independent of the triangle’s real size.

## Part 5. Solved Examples and Practice Problems (With Worksheet)

### Solved Examples

**Question 1.** Find the missing side lengths of a 30 60 90 triangle if the shortest side is 6 units.

- Given: Shortest side (a) = 6 units
- Using the 30 60 90 triangle formulas:
- Medium side (b) = a√3 = 6√3 ≈ 10.39 units
- Longest side (c) = 2a = 2 × 6 = 12 units

**Question 2.** In a 30 60 90 triangle, the hypotenuse is 10 units. Find the lengths of the other two sides.

- Given: Hypotenuse (c) = 10 units
- Using the 30 60 90 triangle formulas:
- Shortest side (a) = c/2 = 10/2 = 5 units
- Medium side (b) = a√3 = 5√3 ≈ 8.66 units

**Question 3.** The medium side of a 30 60 90 triangle is 12 units. Find the lengths of the other two sides.

- Given: Medium side (b) = 12 units
- Using the 30 60 90 triangle formulas:
- Shortest side (a) = b/√3 = 12/√3 ≈ 6.93 units
- Longest side (c) = 2b = 2 × 12 = 24 units

**Question 4.** The shortest side of a 30 60 90 triangle is 8 units. Find the angle measures.

- Given: Shortest side (a) = 8 units
- Using the 30 60 90 triangle rules:
- Angle 1 = 30 degrees
- Angle 2 = 60 degrees
- Angle 3 = 90 degrees

**Question 5.** The longest side of a 30 60 90 triangle is 18 units. Find the lengths of the other two sides.

- Given: Longest side (c) = 18 units
- Using the 30 60 90 triangle formulas:
- Shortest side (a) = c/2 = 18/2 = 9 units
- Medium side (b) = a√3 = 9√3 ≈ 15.59 units

### Practice Problems

### 30 60 90 Triangle Worksheet – Download for free

To further practice working with 30 60 90 triangles, you can download a free worksheet here: [link to worksheet].

## FAQ

### Q1. What is 30-60-90 triangle unit circle？

In trigonometry, the 30 60 90 triangle has tight relationship with the unit circle. Coordinates of the points corresponding to the 30-degree, 60-degree, and 90-degree angles on the unit circle are:

- 30 degrees: (√3/2, 1/2)
- 60 degrees: (1/2, √3/2)
- 90 degrees: (0, 1)

The x-coordinate denotes the adjacent side, the y-coordinate the opposite side, and the hypotenuse is always 1 (the radius of the unit circle), so these coordinates are exactly connected to the side lengths of the 30 60 90 triangle.

### Q2. How do I find the trigonometric ratios in a 30-60-90 triangle?

Angle | Sine | Cosine | Tangent |
---|---|---|---|

30° | 1/2 | √3/2 | 1/√3 |

60° | √3/2 | 1/2 | √3 |

### Q3. What are some practical applications of 30-60-90 triangles?

- Roof construction (the 30-60-90 triangle is often used in the design of roofs)
- Electrical engineering (the 30-60-90 triangle is used in the analysis of three-phase electrical systems)
- Navigation (the 30-60-90 triangle is used in determining direction and distance)

### Q4. How can I use the properties of 30-60-90 triangles to solve problems?

- Identify if a triangle is a 30-60-90 triangle.
- Use the side length relationships to find unknown side lengths.
- Apply the trigonometric ratios to solve for angles or other values.
- Utilize the properties in real-world applications, such as in construction or engineering.

## Conclusion

In geometry and trigonometry, the 30 60 90 triangle is ultimately a particular right triangle with peculiar qualities and formulas that make it significant. Including definition, properties, calculator, rules, formulas, side lengths, and solved examples, this thorough tutorial has addressed the main features of the 30-60-90 triangle.

Understanding the subtleties of this unique triangle can help you to answer many kinds of mathematical problems and apply this knowledge in many spheres, including navigation and beyond as well as building and engineering. The 30 60 90 triangle is a great addition to your mathematical toolkit regardless of your level of geometry knowledge—that, that of a teacher, student, or merely an interested person.

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Get started free!Delvair holds a degree in Physics from the Federal University of Maranhão, Brazil. With over six years of experience, she specializes in teaching mathematics, with a particular emphasis on Math Kangaroo competitions. She firmly believes that education is the cornerstone of society’s future. Additionally, she holds the conviction that every child can learn given the right environment and guidance. In her spare time, she enjoys singing and tending to her plants.

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