Circle Segment Area Calculator

Height (h):

Arc Length (L):

Chord Length (c):

Segment Area (A):

1. What is the Circle Segment Calculator?

Definition: This calculator computes the properties of a circle segment, including its height, arc length, chord length, and area, based on the radius and central angle.

Purpose: It assists in geometry, engineering, and design by providing measurements for circle segments, useful in fields like architecture, optics, and education.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Height (\( h \)): \( h = r - r \times \cos\left(\frac{\alpha}{2}\right) \)
Arc Length (\( L \)): \( L = r \times \alpha \) (where \( \alpha \) is in radians)
Chord Length (\( c \)): \( c = 2 \times r \times \sin\left(\frac{\alpha}{2}\right) \)
Segment Area (\( A \)): \( A = \frac{r^2}{2} \times (\alpha - \sin(\alpha)) \) (where \( \alpha \) is in radians)

Steps:

Enter the radius (r) and central angle (α) in degrees, along with the unit for the radius (mm, cm, m, in, ft, or yd).
Convert the radius to meters and the central angle to radians for calculation.
Compute the height, arc length, chord length, and segment area using the formulas above.
Convert each result to the user-selected unit for that result (lengths to mm, cm, m, in, ft, or yd; area to mm², cm², m², in², ft², or yd²).
Display the results, formatted in scientific notation if the absolute value is less than 0.001, otherwise with 4 decimal places.

3. Importance of Circle Segment Calculations

Calculating the properties of circle segments is essential for:

Architecture:

Optics:

Education:

4. Using the Calculator

Example 1: Calculate the properties of a circle segment:

Radius: \( r = 5 \, \text{in} \);
Central Angle: \( \alpha = 60^\circ \);
Output Units: Height in centimeters, Arc Length in meters, Chord Length in inches, Segment Area in square inches;
Convert to meters: \( r = 5 \times 0.0254 = 0.127 \, \text{m} \), \( \alpha = 60^\circ = \frac{\pi}{3} \, \text{radians} \);
Height: \( h = 0.127 - 0.127 \times \cos\left(\frac{\pi}{6}\right) \approx 0.0169 \, \text{m} \), in centimeters: \( 0.0169 \times 100 \approx 1.69 \, \text{cm} \);
Arc Length: \( L = 0.127 \times \frac{\pi}{3} \approx 0.133 \, \text{m} \), in meters: \( 0.133 \, \text{m} \);
Chord Length: \( c = 2 \times 0.127 \times \sin\left(\frac{\pi}{6}\right) \approx 0.127 \, \text{m} \), in inches: \( 0.127 \div 0.0254 = 5 \, \text{in} \);
Segment Area: \( A = \frac{(0.127)^2}{2} \times \left(\frac{\pi}{3} - \sin\left(\frac{\pi}{3}\right)\right) \approx 0.00212 \, \text{m}^2 \), in in²: \( 0.00212 \div 0.00064516 \approx 3.2859 \, \text{in}^2 \);
Results:
- Height: \( 1.6900 \, \text{cm} \)
- Arc Length: \( 0.1330 \, \text{m} \)
- Chord Length: \( 5.0000 \, \text{in} \)
- Segment Area: \( 3.2859 \, \text{in}^2 \)

5. Frequently Asked Questions (FAQ)

Q: What is a circle segment?
A: A circle segment is the region of a circle bounded by a chord and the arc subtended by the chord. It is defined by the radius and the central angle.

Q: Why is the central angle restricted to 0–360 degrees?
A: A central angle greater than 360 degrees would imply more than one full rotation, which is not a valid segment. The angle must be between 0 and 360 degrees to define a proper segment.

Q: What is the difference between arc length and chord length?
A: The arc length (\( L \)) is the length of the curved boundary of the segment along the circle's circumference. The chord length (\( c \)) is the straight-line distance between the endpoints of the arc.