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 sector formed by the central angle
Area of the triangle formed by the chord and radii
Formula: A = r² arccos((r – h) / r) – (r – h) √(2rh – h²)
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