Find The Major Arc Calculator

Calculate Major Arc Length

Enter the radius of the circle and the central angle of the minor arc to find the length of the major arc.

Major Arc Visualization

Minor Angle (θ)	Minor Arc Length	Major Arc Length
Enter radius and angle to see table

Table showing minor and major arc lengths for various angles with the current radius.

What is a Major Arc?

A major arc is the longer arc connecting two endpoints on a circle. It is the part of the circle’s circumference that is greater than a semicircle. When two points are on a circle, they divide the circle into two arcs: a minor arc (the shorter one) and a major arc (the longer one). The major arc always subtends a reflex angle (an angle greater than 180° and less than 360°) at the center of the circle.

Anyone working with circles in geometry, engineering, design, or even astronomy might need to calculate the length of a major arc. This includes students, architects, engineers, and designers.

A common misconception is that “arc” always refers to the shortest distance along the curve between two points, which is the minor arc. However, the major arc represents the longer path along the circle’s edge between those same two points. Our Major Arc Calculator helps find this longer length.

Major Arc Formula and Mathematical Explanation

To find the length of a major arc, we typically first consider the central angle associated with the minor arc, let’s call it θ (theta), measured in degrees. The central angle associated with the major arc will then be (360 – θ) degrees.

The formula for the length of an arc is derived from the circumference of the circle (2πr). The arc length is a fraction of the circumference, determined by the ratio of the arc’s central angle to the total angle in a circle (360°).

Formula for Minor Arc Length = (θ / 360) * 2 * π * r

Formula for Major Arc Angle = 360° – θ

Formula for Major Arc Length = ((360 – θ) / 360) * 2 * π * r

Alternatively, Major Arc Length = Circumference – Minor Arc Length.

Our Major Arc Calculator uses these formulas.

Variables Table

Variable	Meaning	Unit	Typical Range
r	Radius of the circle	Length units (cm, m, inches, etc.)	r > 0
θ	Central angle of the MINOR arc	Degrees	0 < θ < 360
π (Pi)	Mathematical constant (approx. 3.14159)	Dimensionless	~3.14159
360 – θ	Central angle of the MAJOR arc	Degrees	0 < 360 - θ < 360 (when 0 < θ < 360)

Practical Examples (Real-World Use Cases)

Example 1: Garden Pathway

Imagine a circular garden with a radius of 5 meters. You want to build a paved pathway along the edge, covering a section defined by a minor arc with a central angle of 90 degrees. You want to find the length of the remaining edge (the major arc).

• Radius (r) = 5 m
• Central Angle of Minor Arc (θ) = 90°

Using the Major Arc Calculator or formula:

Minor Arc Length = (90 / 360) * 2 * π * 5 ≈ 7.85 m

Major Arc Angle = 360 – 90 = 270°

Major Arc Length = (270 / 360) * 2 * π * 5 ≈ 23.56 m (or 2*π*5 – 7.85 ≈ 23.56 m)

The longer pathway along the garden edge is about 23.56 meters.

Example 2: Manufacturing a Part

A circular component with a radius of 20 cm needs a slot cut along its edge. The slot corresponds to a minor arc with a central angle of 45°. The remaining edge is the major arc.

• Radius (r) = 20 cm
• Central Angle of Minor Arc (θ) = 45°

Using the Major Arc Calculator:

Minor Arc Length = (45 / 360) * 2 * π * 20 ≈ 15.71 cm

Major Arc Angle = 360 – 45 = 315°

Major Arc Length = (315 / 360) * 2 * π * 20 ≈ 109.96 cm

The length of the component’s edge excluding the slot is about 109.96 cm.

How to Use This Major Arc Calculator

1. Enter the Radius (r): Input the radius of the circle into the “Radius (r)” field. Ensure it’s a positive number. The units (cm, m, inches) will be the same for the arc length.
2. Enter the Central Angle (θ): Input the central angle corresponding to the *minor* arc in degrees into the “Central Angle (θ) of Minor Arc (degrees)” field. This angle must be between 0 and 360 degrees (exclusive of 360 for a defined minor arc, though our calculator handles up to 359.999).
3. Calculate: The calculator automatically updates the results as you type. You can also click the “Calculate” button.
4. Read the Results:
   • Major Arc Length: The primary result, displayed prominently, is the length of the major arc.
   • Intermediate Values: You’ll also see the Minor Arc Length, the Major Arc Angle (360 – θ), and the total Circumference for context.
5. Reset: Click “Reset” to return to the default values.
6. Copy: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

Understanding the major arc length is crucial when you need to know the length of the longer segment of a circle’s edge between two points.

Key Factors That Affect Major Arc Length Results

1. Radius (r): The larger the radius, the larger the circumference, and thus the longer both the minor and major arc lengths will be for a given angle. It directly scales the arc length.
2. Central Angle of Minor Arc (θ): As the central angle of the minor arc increases, the minor arc length increases, and consequently, the major arc length decreases (since the major arc angle 360-θ decreases). Conversely, a smaller minor arc angle results in a longer major arc.
3. Unit of Radius: The unit used for the radius (cm, m, inches, etc.) will be the unit for the calculated arc lengths. Ensure consistency.
4. Accuracy of π: The value of π used in the calculation affects precision. Our Major Arc Calculator uses `Math.PI` for good accuracy.
5. Angle Measurement: The angle must be in degrees for the formula (θ/360) to be correct. If your angle is in radians, convert it first (Degrees = Radians * 180/π).
6. Relationship to Minor Arc: The major arc and minor arc together make up the full circumference. Knowing one and the circumference allows you to find the other.

Frequently Asked Questions (FAQ)

What is the difference between a major arc and a minor arc?

A minor arc is the shorter arc connecting two points on a circle’s circumference, while a major arc is the longer arc connecting the same two points. The minor arc’s central angle is less than 180°, and the major arc‘s central angle (reflex angle) is greater than 180°.

How is the major arc measured?

The major arc can be measured by its length (like our calculator does) or by the measure of its central angle (which is 360° minus the central angle of the minor arc).

What units are used for the major arc length?

The units of the major arc length will be the same as the units used for the radius (e.g., cm, meters, inches).

Can the central angle of the minor arc be greater than 180 degrees?

By definition, the minor arc is the shorter one, so its central angle is usually considered to be less than 180°. If the angle given is between 180° and 360°, it technically defines the major arc directly, and the arc with angle 360-θ would be the minor one. Our calculator assumes the input angle is for the arc you are initially defining as ‘minor’ relative to the major arc you want to find.

What if the two points are at opposite ends of a diameter?

If the two points are diametrically opposite, they divide the circle into two semicircles, each with a central angle of 180°. In this case, there isn’t a distinct major or minor arc; both arcs are equal (semicircles).

Can I find the major arc if I only know the minor arc length and radius?

Yes. First calculate the circumference (C = 2πr). Then, the Major Arc Length = C – Minor Arc Length.

Does the Major Arc Calculator work with radians?

This calculator specifically uses degrees for the input angle. If you have the angle in radians, convert it to degrees first (Degrees = Radians * 180/π) before using the calculator.

What is a reflex angle in the context of a major arc?

The central angle subtended by the major arc at the center of the circle is always a reflex angle, meaning it is greater than 180° and less than 360°.

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