Understanding the Calculation of the Area of a Circular Segment

The calculation of the area of a circular segment is essential in geometry and engineering. It involves determining the region bounded by a chord and the corresponding arc of a circle.

This article explores detailed formulas, common values, and real-world applications for accurately computing circular segment areas. Readers will gain expert-level insights and practical examples.

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Calculate the area of a circular segment with radius 10 cm and chord length 12 cm.
Find the segment area when the central angle is 60 degrees and radius is 8 meters.
Determine the area of a segment with radius 15 inches and segment height 4 inches.
Compute the circular segment area for a radius of 5 m and chord length of 6 m.

Comprehensive Tables of Common Circular Segment Values

Below are extensive tables showing the area of circular segments for various common radii, chord lengths, segment heights, and central angles. These tables serve as quick references for engineers, architects, and mathematicians.

Radius (r)	Chord Length (c)	Segment Height (h)	Central Angle (θ in degrees)	Area of Segment (A)
5	6	1.2	70.53	4.18
5	8	2.0	106.26	8.14
10	12	2.4	70.53	16.75
10	16	4.0	106.26	32.56
15	18	3.6	70.53	37.65
15	24	6.0	106.26	73.27
20	24	3.6	43.59	22.44
20	28	5.0	57.12	33.49
25	30	3.6	34.85	28.14
25	40	8.0	73.74	97.43

Note: Radius (r), chord length (c), and segment height (h) are in consistent units (cm, m, inches). Area (A) is in square units corresponding to the radius unit.

Fundamental Formulas for Calculating the Area of a Circular Segment

The area of a circular segment can be calculated using several interrelated formulas depending on the known parameters: radius, chord length, segment height, or central angle. Below are the primary formulas with detailed explanations.

1. Area Using Radius and Central Angle

The most direct formula uses the radius r and the central angle θ (in radians) subtended by the segment’s arc:

A = (r2 / 2) × (θ – sin(θ))

A: Area of the circular segment
r: Radius of the circle
θ: Central angle in radians

This formula calculates the difference between the area of the sector and the area of the triangular portion formed by the chord and the circle’s center.

2. Area Using Radius and Segment Height

If the segment height h (the perpendicular distance from the chord to the arc) is known, the area can be computed as:

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

h: Segment height
arccos: Inverse cosine function (returns radians)

This formula derives from subtracting the triangular area under the chord from the sector area defined by the central angle.

3. Area Using Radius and Chord Length

When the chord length c is known, first calculate the segment height h:

h = r – √(r2 – (c / 2)2)

Then use the formula for area with radius and height as above.

c: Chord length

4. Calculating the Central Angle from Radius and Chord Length

The central angle θ in radians can be found by:

θ = 2 × arccos((r – h) / r) = 2 × arcsin(c / (2r))

This is useful when only the chord length and radius are known.

Explanation of Variables and Typical Values

Radius (r): The distance from the center of the circle to any point on its circumference. Common engineering radii range from millimeters to meters depending on application.
Chord Length (c): The straight-line distance between two points on the circle’s circumference. It is always less than or equal to the diameter (2r).
Segment Height (h): The perpendicular distance from the chord to the arc. It ranges from 0 (chord coincides with arc) to r (semicircle).
Central Angle (θ): The angle subtended at the center of the circle by the arc. It ranges from 0 to 2π radians (0° to 360°).

Real-World Applications and Detailed Examples

Understanding the area of a circular segment is critical in various fields such as civil engineering, architecture, mechanical design, and optics. Below are two detailed real-world examples demonstrating the calculation process.

Example 1: Designing a Curved Window Arch

An architect is designing a semicircular window with a radius of 2 meters. The window includes a horizontal chord 3 meters long at the base. The architect needs to calculate the area of the glass segment above the chord to estimate material requirements.

Step 1: Identify Known Values

Radius, r = 2 m
Chord length, c = 3 m

Step 2: Calculate Segment Height (h)

h = r – √(r2 – (c / 2)2) = 2 – √(4 – (1.5)2) = 2 – √(4 – 2.25) = 2 – √1.75 ≈ 2 – 1.322 = 0.678 m

Step 3: Calculate Central Angle (θ)

θ = 2 × arccos((r – h) / r) = 2 × arccos((2 – 0.678) / 2) = 2 × arccos(0.661) ≈ 2 × 0.851 = 1.702 radians

Step 4: Calculate Area of Segment (A)

A = (r2 / 2) × (θ – sin(θ)) = (4 / 2) × (1.702 – sin(1.702)) = 2 × (1.702 – 0.991) = 2 × 0.711 = 1.422 m2

Result: The glass area required for the segment is approximately 1.422 square meters.

Example 2: Hydraulic Engineering – Water Flow Under a Circular Arch

In hydraulic engineering, calculating the cross-sectional area of water flowing under a circular arch is essential. Suppose a culvert has a circular arch with radius 3 meters. The water surface forms a chord 4 meters wide. Determine the cross-sectional area of water flow.

Step 1: Known Parameters

Radius, r = 3 m
Chord length, c = 4 m

Step 2: Calculate Segment Height (h)

h = r – √(r2 – (c / 2)2) = 3 – √(9 – 22) = 3 – √(9 – 4) = 3 – √5 ≈ 3 – 2.236 = 0.764 m

Step 3: Calculate Central Angle (θ)

θ = 2 × arccos((r – h) / r) = 2 × arccos((3 – 0.764) / 3) = 2 × arccos(0.745) ≈ 2 × 0.732 = 1.464 radians

Step 4: Calculate Area of Segment (A)

A = (r2 / 2) × (θ – sin(θ)) = (9 / 2) × (1.464 – sin(1.464)) = 4.5 × (1.464 – 0.994) = 4.5 × 0.470 = 2.115 m2

Result: The cross-sectional area of water flow under the arch is approximately 2.115 square meters.

Additional Considerations and Advanced Insights

For highly precise engineering applications, consider the following:

Unit Consistency: Always ensure radius, chord length, and height are in the same units to avoid calculation errors.
Angle Units: Convert degrees to radians when using trigonometric functions: radians = degrees × (π / 180).
Numerical Methods: For complex shapes or when parameters are partially unknown, numerical integration or CAD software may be used to approximate segment areas.
Material Properties: In structural applications, the segment area may relate to load distribution, requiring integration with stress analysis.

For further reading on circular geometry and segment calculations, authoritative resources include:

Summary of Key Formulas

Area from radius and central angle:

A = (r2 / 2) × (θ – sin(θ))
  Area from radius and segment height:

A = r2 × arccos((r – h) / r) – (r – h) × √(2rh – h2)
  Segment height from radius and chord length:

h = r – √(r2 – (c / 2)2)
  Central angle from radius and chord length:

θ = 2 × arccos((r – h) / r) = 2 × arcsin(c / (2r))

Mastering these formulas enables precise calculation of circular segment areas for diverse technical applications.