A **circle segment calculator** is a **mathematical tool** used to determine various properties of a **portion of a circle** cut off by a **chord**.

This segment is formed when a **straight line** intersects a circle, creating two distinct parts. The calculator typically helps in finding the **area**, **height**, and **length of the arc** of the segment.

**Examples:**

**Calculating**the area of a**slice of pizza****Determining**the volume of a**partially filled cylindrical tank****Estimating**the material needed for a**curved architectural feature**

## Circle Segment Calculator

Radius (r) | Height (h) | Chord Length (c) | Area (A) | Arc Length (L) | Central Angle (θ) |
---|---|---|---|---|---|

10 cm | 3 cm | 9.17 cm | 40.18 cm² | 11.22 cm | 1.29 rad (73.74°) |

5 m | 1 m | 3.87 m | 6.59 m² | 4.19 m | 0.84 rad (48.19°) |

15 ft | 4 ft | 13.42 ft | 80.91 ft² | 16.97 ft | 1.13 rad (64.62°) |

20 in | 6 in | 18.33 in | 161.17 in² | 23.09 in | 1.15 rad (66.04°) |

**Conversion Equations:**

**Radians to Degrees:****θ(degrees) = θ(radians) * (180/π)****Area Conversion:****1 m² = 10.7639 ft² = 1550.0031 in²**

## Circle Segment Formula

The primary formula for a circle segment involves calculating its **area**:

**A = r² arccos((r – h) / r) – (r – h) √(2rh – h²)**

Where:

**A**: Area of the segment**r**: Radius of the circle**h**: Height of the segment

**Example:**

For a circle with a **radius** of 10 units and segment **height** of 3 units, the area would be approximately **40.18 square units**.

## How to Calculate a Segment of a Circle

**Measure the radius**(**r**) of the circle**Determine the height**(**h**) of the segment**Calculate the central angle**(**θ**) using the formula:**θ = 2 * arccos((r – h) / r)****Find the arc length**using:**L = r * θ****Compute the area**using the formula mentioned earlier

This **step-by-step process** ensures accurate calculations for various segment properties.

## How do you find the depth of a circle?

To find it:

- Measure the
**chord length**(**c**) and**radius**(**r**) - Use the formula:
**h = r – √(r² – (c/2)²)**

The **depth** or **height** of a circle segment is the **perpendicular distance** from the chord to the circle’s circumference.

This calculation is crucial for **engineering applications**, such as determining **fluid levels** in tanks.

## What is the formula for segment height?

The segment height formula can be rearranged to solve for different variables:

**h = r – √(r² – (c/2)²)**

Where:

**h**: Height of the segment**r**: Radius of the circle**c**: Chord length

This **versatile formula** allows for quick calculations when certain dimensions are known.

## How to calculate partial circle area?

To find the area of a **partial circle** (sector):

**Calculate the central angle** (**θ**) in **radians**

Use the formula: **A = (1/2) r² θ**

This method is useful for sectors that aren’t complete segments, like in **pie charts** or **gear designs**.

## Area of a Circle Segment

The area of a circle segment combines two calculations:

Area of the sectorformed by the central angle

Area of the triangleformed by the chord and radii

**Formula:** **A = r² arccos((r – h) / r) – (r – h) √(2rh – h²)**

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