Circular Segment Area Calculation

Defining a Circular Segment

A circular segment is a region bounded by a chord and the arc of a circle. A minor segment is the smaller of the two areas created by a chord dividing a circle.

Calculating the Area

Method 1: Using the Radius and Central Angle

The area (A) of a minor segment can be calculated using the radius (r) of the circle and the central angle (θ, in radians) subtended by the arc:

A = (1/2)r²(θ - sinθ)

Method 2: Using the Radius and Chord Length

Alternatively, if the chord length (c) is known instead of the central angle, the area can be calculated using the following steps:

1. Calculate the height (h) of the segment: h = r - (r² - (c/2)²)^1/2
2. Calculate the central angle (θ) using the formula: θ = 2 arcsin(c / (2r))
3. Calculate the area (A) using the formula from Method 1: A = (1/2)r²(θ - sinθ)

Method 3: Approximating the Area with a Triangle

For small segments, where the arc closely approximates a straight line, the area can be approximated using the area of a triangle. This is generally less accurate than the above methods.

Important Considerations

• Ensure that the angle θ is in radians when using the formulas. Convert degrees to radians using the conversion factor π/180.
• The accuracy of the approximation method decreases as the segment size increases.
• For calculations, using appropriate units (e.g., meters for radius, radians for angle) is crucial for obtaining correct results.

Applications

Calculating the area of a circular segment has applications in various fields, including engineering, surveying, and computer graphics.