To calculate the area enclosed by a polar curve, use the formula for the area of a polar curve by integrating over the desired angle range. This helps you find the total area within the polar region.

## Polar Area Calculator

Enter any 2 values to calculate the missing variable

The **Polar Area Calculator** is a helpful tool for determining the area enclosed by a polar curve, or the area between two polar curves. Polar coordinates are often used to describe shapes with circular symmetry, and calculating areas in this system requires specialized formulas. The calculator simplifies this process, allowing you to input polar equations and angle ranges to compute the enclosed area accurately. This is useful for visualizing polar graphs, finding areas under curves, and solving math problems involving polar coordinates.

### Formula:

$A = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left( r(\theta) \right)^2 d\theta$

Variable |
Description |
---|---|

A |
Area enclosed by the polar curve |

r(θ) |
Polar function, describing the distance from the origin |

θ₁, θ₂ |
Angle range (in radians) for which the area is calculated |

### Solved Calculation:

**Example 1:**

Step |
Calculation |
---|---|

Polar function (r(θ)) | r(θ) = 2 + sin(θ) |

Angle range (θ₁ to θ₂) | 0 to π |

Area Calculation |
A = ½ ∫(0 to π) (2 + sin(θ))² dθ |

Result |
7.858 units² |

**Answer**: The area enclosed by the polar curve from 0 to π is approximately 7.858 units².

**Example 2:**

Step |
Calculation |
---|---|

Polar function (r(θ)) | r(θ) = 3sin(θ) |

Angle range (θ₁ to θ₂) | 0 to π/2 |

Area Calculation |
A = ½ ∫(0 to π/2) (3sin(θ))² dθ |

Result |
3.534 units² |

**Answer**: The area enclosed by the polar curve from 0 to π/2 is approximately 3.534 units².