Circular Segment Area Calculator | Calculate Area from Radius, Angle, Chord

What is a Circular Segment?

Defining the Geometry
Key Components of a Segment
Minor vs. Major Segments

A circular segment is a region of a circle that is 'cut off' from the rest of the circle by a straight line called a chord. Imagine slicing a pizza not from the center outwards, but straight across. The smaller piece you get is a circular segment. This area is bounded by the chord and the arc of the circle.

Defining the Geometry

To precisely define a segment, you need to know the properties of the circle it belongs to, specifically its radius. The segment itself is then defined by either the length of its chord or the central angle it subtends.

Key Components of a Segment

• Chord (c): The straight line that connects two points on the circle's circumference, forming the base of the segment.

• Arc: The curved part of the circle's circumference that forms the top boundary of the segment.

• Radius (R): The radius of the full circle from which the segment is cut.

• Central Angle (θ): The angle formed at the center of the circle by two radii that connect to the endpoints of the chord.

• Height (h): The perpendicular distance from the midpoint of the chord to the arc. This is also known as the sagitta.

Minor vs. Major Segments

A single chord divides a circle into two segments. The smaller segment is called the minor segment, and the larger one is the major segment. The minor segment has a central angle less than 180° (π radians), while the major segment has a central angle greater than 180°.

Visualizing Segments

A round window with a flat bottom sill is a classic example of a circular segment.
The cross-section of a partially filled horizontal pipe contains a circular segment of liquid.

Step-by-Step Guide to Using the Calculator

Choosing Your Calculation Method
Entering Input Values
Interpreting the Results

Our calculator is designed for ease of use. Here's how to get your results in just a few steps.

Choosing Your Calculation Method

Start by selecting your preferred calculation method from the dropdown menu. You have two options:

• From Radius and Central Angle: Use this if you know the circle's radius and the central angle (in degrees or radians) that corresponds to the segment.

• From Radius and Chord Length: Use this if you know the radius and the length of the straight-line chord that defines the segment.

Entering Input Values

Once you've selected a method, input fields for the required parameters will appear. Enter your values into the designated boxes. The tool has built-in validation to ensure the numbers are logical (e.g., the chord length cannot exceed the circle's diameter).

Interpreting the Results

After clicking 'Calculate', the tool will instantly display the results. This includes not only the Segment Area but also other useful geometric properties like the Arc Length and the Chord Length (if you calculated from an angle). These results are essential for further engineering or design calculations.

Calculation Scenarios

If you're designing an arched doorway, you might know the radius of the arch and the width of the door (chord length).
In a physics problem, you might be given the angle of a pendulum's swing and its length (radius) to find the area it covers.

Key Formulas and Mathematical Derivation

The Core Concept: Sector Minus Triangle
Formula 1: Using Radius and Central Angle
Formula 2: Deriving from Radius and Chord

The area of a circular segment is found by calculating the area of the circular sector formed by the central angle and then subtracting the area of the isosceles triangle formed by the radii and the chord.

The Core Concept: Sector Minus Triangle

AreaSegment = AreaSector - Area_Triangle

Formula 1: Using Radius and Central Angle (θ)

This is the most direct method. The formulas are:

• Area of Sector: A_sector = (θ / 2) * R² (where θ is in radians)

• Area of Triangle: A_triangle = (1/2) R² sin(θ)

• Area of Segment: A_segment = (1/2) R² (θ - sin(θ))

If your angle is in degrees, you must first convert it to radians: Radians = Degrees * (π / 180).

Formula 2: Deriving from Radius (R) and Chord (c)

When you know the chord length instead of the angle, you first need to find the central angle θ.

• Find the half angle: The radius and half the chord (c/2) form a right-angled triangle. So, sin(θ/2) = (c/2) / R.

• Calculate the full angle: θ = 2 * arcsin(c / (2R)).

• Apply the main formula: Once you have θ in radians, you can plug it into the segment area formula: A = (1/2) R² (θ - sin(θ)).

Numerical Example

Given R=10, c=12. First, find θ = 2 * arcsin(12 / (2*10)) = 2 * arcsin(0.6) ≈ 1.287 radians.
Then, Area = 0.5 * 10² * (1.287 - sin(1.287)) = 50 * (1.287 - 0.963) ≈ 16.2 square units.

Real-World Applications of Segment Area

Architecture and Construction
Engineering and Fluid Dynamics
Agriculture and Land Measurement

The calculation of a circular segment's area is not just an academic exercise; it has numerous practical applications across various fields.

Architecture and Construction

Designers and architects frequently use this calculation for features like arched windows, doorways, and vaulted ceilings. Determining the area is crucial for estimating material costs (like glass or stone) and for structural analysis.

Engineering and Fluid Dynamics

Engineers often need to calculate the volume of liquid in a partially filled horizontal cylindrical tank. The cross-sectional area of the liquid is a circular segment. This calculation is vital for managing fluid levels in tanks, pipes, and channels.

Agriculture and Land Measurement

When measuring irregularly shaped plots of land, especially those bounded by curved features like rivers or roads, surveyors may approximate sections of the land as circular segments to calculate the total area for farming or valuation.

Application Scenarios

Calculating the amount of paint needed for a semi-circular gable.
Determining the fuel level in a horizontal cylindrical fuel tanker based on the liquid's height.

Common Questions and Advanced Topics

What if the angle is greater than 180°?
How is segment height related?
Limitations and Accuracy

Let's address some common questions and delve into more advanced aspects of circular segments.

What if the angle is greater than 180° (a Major Segment)?

The formulas work perfectly for major segments as well. An angle greater than 180° (π radians) will result in a larger segment area. Alternatively, you can calculate the area of the corresponding minor segment and subtract it from the total area of the circle (A_circle = πR²).

How is segment height (h) related?

The height (h) or sagitta of a segment is also a key parameter. While this calculator doesn't use height as a primary input, it's related to the radius and central angle by the formula: R - h = R * cos(θ/2). Knowing any two parameters (R, c, h, θ) allows you to derive the others.

Limitations and Accuracy

The accuracy of the calculation depends on the precision of your input values. When dealing with physical measurements, ensure they are as accurate as possible. The calculations in this tool use floating-point arithmetic, providing a high degree of precision suitable for most academic and professional applications.

Advanced Thoughts

For a semicircle, the central angle is 180° (π radians), and the segment area is exactly half the circle's area.
If the chord length is equal to the diameter (2R), the segment is the entire circle, but this scenario is often handled as a full circle calculation.

Example 1: Using Radius and Angle in Degrees

Example 2: Using Radius and Angle in Radians

Example 3: Using Radius and Chord Length

Example 4: Large Angle Segment (Major Segment)