A Triangle Rotation Calculator is an invaluable tool designed to simplify the complex process of rotating triangles, whether in a plane around the origin or around any arbitrary point. This functionality is essential in various fields such as computer graphics, engineering, and mathematics, where precise rotation calculations are required to manipulate shapes accurately. By inputting the coordinates of the triangle’s vertices and the angle of rotation, users can instantly obtain the new coordinates of the rotated triangle, thereby facilitating efficient and error-free transformations.

## Formula of Triangle Rotation Calculator

#### Rotation – Around the Origin:

When rotating a triangle around the origin (0,0) by an angle Θ (theta), the transformation of each vertex (x,y) follows specific formulas:

```
New x coordinate (x'): x' = x * cos(Θ) - y * sin(Θ)
New y coordinate (y'): y' = x * sin(Θ) + y * cos(Θ)
```

#### Rotation – Around an Arbitrary Point:

For rotations around a different point (xo, yo) by an angle Θ, the formulas adapt as follows:

```
New x coordinate (x'): x' = xo + (x - xo) * cos(Θ) - (y - yo) * sin(Θ)
New y coordinate (y'): y' = yo + (x - xo) * sin(Θ) + (y - yo) * cos(Θ)
```

These equations consider Θ as the rotation angle, which can be input in degrees or radians depending on your calculator’s settings. The coordinates (x, y) represent the original positions of each vertex before rotation.

## General Terms Table

Angle (Degrees) | Angle (Radians) | Cosine | Sine |
---|---|---|---|

0 | 0 | 1 | 0 |

30 | π/6 | √3/2 | 1/2 |

45 | π/4 | √2/2 | √2/2 |

60 | π/3 | 1/2 | √3/2 |

90 | π/2 | 0 | 1 |

180 | π | -1 | 0 |

This table provides a quick reference for common angles in both degrees and radians, along with their cosine and sine values, facilitating easy calculations without the need for manual conversions or computations.

## Example of Triangle Rotation Calculator

**Given:** A triangle with vertices A(1, 2), B(3, 4), and C(5, 6) to be rotated 90 degrees around the origin.

**Rotation Formulas for 90 Degrees (π/2 Radians):**

- Cos(90°) = 0, Sin(90°) = 1

**Calculating New Coordinates:**

Forvertex A(1, 2):

- New x = 1 * 0 – 2 * 1 = -2
- New y = 1 * 1 + 2 * 0 = 1
**New A'(-2, 1)**

For vertex B(3, 4):

- New x = 3 * 0 – 4 * 1 = -4
- New y = 3 * 1 + 4 * 0 = 3
**New B'(-4, 3)**

For vertex C(5, 6):

- New x = 5 * 0 – 6 * 1 = -6
- New y = 5 * 1 + 6 * 0 = 5
**New C'(-6, 5)**

**Result:** After rotating 90 degrees around the origin, the new vertices of the triangle are A'(-2, 1), B'(-4, 3), and C'(-6, 5).

## Most Common FAQs

**What is the Triangle Rotation Calculator use for?**

The Triangle Rotation Calculator is a tool design to compute the new coordinates of a triangle’s vertices after it has been rotate by a specific angle around the origin or any arbitrary point. It is widely use in fields requiring geometric transformations, such as computer graphics and engineering.

**Can the calculator handle rotations in both degrees and radians?**

Yes, the calculator can process rotation angles in both degrees and radians. Users should ensure they input the angle in the correct unit based on their calculator settings.

**How do I use the calculator for rotations around a point other than the origin?**

To rotate a triangle around a point other than the origin, input the coordinates of that point along with the triangle’s vertices and the desired angle of rotation. The calculator’s formula adjusts to account for this point, providing accurate results for any pivot point.