Find Segment Of Circle Calculator

Calculate Segment Area

Enter the radius of the circle and the central angle (in degrees) subtended by the segment to calculate the segment’s area and other properties.

Segment Area for Different Angles

Angle (θ)	Segment Area	Sector Area	Triangle Area

Table showing how the segment area varies with the angle for the entered radius.

What is a Segment of a Circle?

A segment of a circle is the region bounded by a chord and the arc subtended by that chord. Imagine cutting a slice of pizza – if you cut straight from one point on the edge to another (not through the center), the smaller piece you cut off is a segment. The larger part is also a segment.

A segment is formed by a chord and an arc. The area of the segment is the area of the sector formed by the two radii to the ends of the chord, minus the area of the isosceles triangle formed by the two radii and the chord.

Anyone studying geometry, engineering, design, or architecture might need to use a find segment of circle calculator. It’s useful in various fields where circular shapes and their parts are analyzed.

A common misconception is confusing a segment with a sector. A sector is like a slice of pizza cut from the center, bounded by two radii and an arc. A segment is bounded by a chord and an arc.

Segment of Circle Formula and Mathematical Explanation

To find the area of a circular segment, we first calculate the area of the circular sector corresponding to the central angle θ, and then subtract the area of the isosceles triangle formed by the two radii and the chord.

1. Convert Angle to Radians: If the angle θ is given in degrees, convert it to radians: θ_rad = θ * (π / 180).

2. Area of the Sector: The area of a sector with radius ‘r’ and central angle θ_rad (in radians) is:
Area_sector = (1/2) * r² * θ_rad

3. Area of the Triangle: The area of the isosceles triangle formed by the two radii and the chord is:
Area_triangle = (1/2) * r² * sin(θ_rad)

4. Area of the Segment: The area of the segment is the difference between the sector area and the triangle area:
Area_segment = Area_sector – Area_triangle = (1/2) * r² * (θ_rad – sin(θ_rad))

When θ is in degrees, the formula becomes:

Area_segment = (1/2) * r² * ( (θ * π / 180) – sin(θ * π / 180) )

Or, simplifying: Area_segment = r² * ( (θ * π / 360) – (sin(θ * π / 180) / 2) )

Variables Table

Variable	Meaning	Unit	Typical Range
r	Radius of the circle	Length (e.g., cm, m, inches)	> 0
θ	Central angle in degrees	Degrees	0 < θ ≤ 360
θrad	Central angle in radians	Radians	0 < θrad ≤ 2π
Areasegment	Area of the circular segment	Area (e.g., cm², m², inches²)	≥ 0
Areasector	Area of the circular sector	Area	≥ 0
Areatriangle	Area of the isosceles triangle	Area	≥ 0
c	Chord length	Length	0 < c ≤ 2r
s	Arc length	Length	0 < s ≤ 2πr
h	Segment height (sagitta)	Length	0 ≤ h ≤ r (for θ ≤ 180) or r < h <= 2r (for θ > 180)

Practical Examples (Real-World Use Cases)

Example 1: Designing a Window

An architect is designing a window with a circular top. The circular part has a radius of 50 cm, and the segment base (chord) is formed at a central angle of 120 degrees.

• Radius (r) = 50 cm
• Angle (θ) = 120 degrees

Using the find segment of circle calculator or the formula:
Angle in radians = 120 * π / 180 = 2π/3 ≈ 2.094 radians.
Area of sector = 0.5 * 50² * 2.094 ≈ 2617.5 cm².
Area of triangle = 0.5 * 50² * sin(2.094) ≈ 0.5 * 2500 * 0.866 = 1082.5 cm².
Area of segment = 2617.5 – 1082.5 = 1535 cm².

The area of the glass needed for the segment is 1535 cm².

Example 2: Landscaping

A gardener is creating a flower bed in the shape of a segment of a circle with a radius of 3 meters and a central angle of 90 degrees.

• Radius (r) = 3 m
• Angle (θ) = 90 degrees

Angle in radians = 90 * π / 180 = π/2 ≈ 1.571 radians.
Area of sector = 0.5 * 3² * 1.571 = 0.5 * 9 * 1.571 ≈ 7.0695 m².
Area of triangle = 0.5 * 3² * sin(1.571) = 0.5 * 9 * 1 = 4.5 m².
Area of segment = 7.0695 – 4.5 = 2.5695 m².

The area of the flower bed is approximately 2.57 m².

How to Use This Find Segment of Circle Calculator

Our find segment of circle calculator is straightforward:

1. Enter Radius (r): Input the radius of the circle in the first field. Ensure it’s a positive number.
2. Enter Central Angle (θ): Input the central angle in degrees (between 0 and 360) in the second field.
3. View Results: The calculator automatically updates and displays the Segment Area (primary result), along with Sector Area, Triangle Area, Chord Length, Arc Length, and Segment Height. The formula used is also shown.
4. See Visuals: The chart and table update based on your inputs to give you a visual understanding and comparative data.
5. Reset: Click “Reset” to return to default values.
6. Copy: Click “Copy Results” to copy the main results and inputs to your clipboard.

The results help you understand the area and dimensions of the segment. If you’re planning material usage or design, these values are crucial.

Key Factors That Affect Segment of Circle Results

The area and other properties of a circular segment are primarily affected by:

1. Radius (r): The larger the radius, the larger the segment area for a given angle, as the area scales with r².
2. Central Angle (θ): The area of the segment increases as the angle increases from 0 to 180 degrees. Beyond 180 degrees, you’re usually looking at the major segment, and the minor segment’s area starts decreasing as the angle approaches 360 (or 0 again). The relationship is not linear due to the sine function.
3. Angle Unit: Ensure the angle is correctly interpreted (degrees in our calculator). Using radians where degrees are expected will give vastly different results.
4. Chord Length (c): While not a direct input in this version of the calculator (it’s calculated), if you know the chord length and radius, you can find the angle, which then affects the area. A longer chord (for a given radius) means a larger angle and usually a larger segment area (up to 180 degrees).
5. Segment Height (h): Similar to chord length, the height (sagitta) is related to the angle and radius and influences the area.
6. Calculation Precision: The value of π and the precision of the sine function used in calculations affect the final result’s accuracy. Our find segment of circle calculator uses standard `Math.PI` and `Math.sin` for good precision.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a sector and a segment of a circle?

A sector is a region bounded by two radii and an arc (like a pizza slice). A segment is a region bounded by a chord and an arc.

Q2: How do I find the area of a major segment?

If you calculate the area of the minor segment (using an angle less than 180 degrees), the area of the major segment is the total area of the circle (πr²) minus the area of the minor segment.

Q3: Can the central angle be greater than 180 degrees for a segment?

Yes. If the angle is greater than 180 degrees, you are typically defining the major segment through that angle, or the minor segment is defined by (360 – θ). Our find segment of circle calculator handles angles up to 360, calculating the area bounded by the arc defined by that angle.

Q4: What if I know the chord length and radius, but not the angle?

You can find the half-angle (θ/2) using sin(θ/2) = (c/2)/r, where c is the chord length. Then find θ and use the segment area formula. We might add a chord length calculator feature later.

Q5: What is the segment height or sagitta?

The segment height (h), or sagitta, is the distance from the midpoint of the chord to the arc, measured along the radius that bisects the chord. h = r * (1 – cos(θ/2)).

Q6: Is the area of the segment ever negative?

No, the area should always be positive. The formula (θ_rad – sin(θ_rad)) is always non-negative for 0 ≤ θ_rad ≤ 2π.

Q7: How accurate is this find segment of circle calculator?

It uses standard JavaScript Math functions, which provide good precision for most practical purposes.

Q8: What are real-world applications of calculating a circle segment area?

Applications include architecture (arched windows), engineering (pipe cross-sections with fluid), design (logos, patterns), and landscaping (curved flower beds). Our area calculator covers various shapes.

Related Tools and Internal Resources

• Area Calculator: Calculate areas of various geometric shapes, including circles and triangles.
• Circle Calculator: Find circumference, area, and diameter of a circle given one parameter.
• Geometry Formulas: A collection of common geometry formulas.
• Arc Length Calculator: Calculate the length of a circular arc.
• Chord Length Calculator: Find the length of a chord given radius and angle.
• Sector Area Calculator: Specifically calculate the area of a circular sector.