Find Radius Of Circle With Arc Length Calculator

Easily determine the radius of a circle given its arc length and the central angle subtended by the arc using our radius of circle with arc length calculator.

Calculate Radius from Arc Length

Results copied!

Radius for different Arc Lengths and a Fixed Angle (90°)

Arc Length (L)	Angle (θ)	Radius (r)
5	90°	3.18
10	90°	6.37
15	90°	9.55
20	90°	12.73

What is a Radius of Circle with Arc Length Calculator?

A radius of circle with arc length calculator is a tool used to determine the radius of a circle when you know the length of an arc on that circle and the central angle that subtends (forms) that arc. In geometry, an arc is a portion of the circumference of a circle, and the central angle is the angle whose vertex is the center of the circle and whose sides pass through the endpoints of the arc.

This calculator is particularly useful for students, engineers, designers, and anyone working with circular or arc-based measurements. If you have the arc length and the angle, the radius of circle with arc length calculator saves you from manual calculations.

Common misconceptions include thinking the arc length is the straight-line distance between the arc's endpoints (that's the chord length) or confusing the central angle with an inscribed angle.

Radius of Circle with Arc Length Formula and Mathematical Explanation

The relationship between the arc length (L), the radius (r), and the central angle (θ) is fundamental in geometry. The formula depends on whether the angle is measured in degrees or radians.

When the Angle is in Radians:

The simplest form of the relationship is when the angle θ is measured in radians. The arc length is directly proportional to the radius and the angle:

L = r * θ

From this, we can derive the formula for the radius:

r = L / θ (where θ is in radians)

When the Angle is in Degrees:

If the angle θ is measured in degrees, we first need to convert it to radians because the direct formula `L = r * θ` requires θ in radians. To convert degrees to radians, we use the conversion factor π / 180:

Angle in radians = Angle in degrees * (π / 180)

So, L = r * (Angle in degrees * π / 180)

Rearranging to find the radius (r):

r = (L * 180) / (Angle in degrees * π)

Our radius of circle with arc length calculator uses these formulas based on the unit you select for the angle.

Variables Table:

Variable	Meaning	Unit	Typical Range
L	Arc Length	Any unit of length (e.g., cm, m, inches)	> 0
θ	Central Angle	Degrees (°) or Radians (rad)	> 0 (usually up to 360° or 2π rad)
r	Radius	Same unit as L	> 0
π	Pi	Dimensionless	~3.14159

Variables used in the radius of circle with arc length calculator.

Practical Examples (Real-World Use Cases)

Let's see how the radius of circle with arc length calculator can be used in real scenarios.

Example 1: Designing a Curved Path

Imagine an architect is designing a curved garden path. They want the path (arc) to be 15 meters long, and it should curve through an angle of 60 degrees. What is the radius of the circle that this path is part of?

• Arc Length (L) = 15 m
• Angle (θ) = 60 degrees

Using the formula r = (L * 180) / (θ * π):

r = (15 * 180) / (60 * π) = 2700 / (60 * 3.14159) ≈ 2700 / 188.4954 ≈ 14.32 meters

The radius of the curve is approximately 14.32 meters.

Example 2: Engineering a Pipe Bend

An engineer is designing a pipe that needs to bend. The bent section (arc) has a length of 2 feet, and the angle of the bend is 1.5 radians. What is the radius of the bend?

• Arc Length (L) = 2 feet
• Angle (θ) = 1.5 radians

Using the formula r = L / θ (since the angle is in radians):

r = 2 / 1.5 ≈ 1.33 feet

The radius of the pipe bend is approximately 1.33 feet. Our radius of circle with arc length calculator can quickly give these results.

How to Use This Radius of Circle with Arc Length Calculator

Using our radius of circle with arc length calculator is straightforward:

1. Enter Arc Length (L): Input the known length of the arc into the "Arc Length (L)" field. Make sure it's a positive number.
2. Enter Angle (θ): Input the value of the central angle subtended by the arc in the "Angle (θ)" field.
3. Select Angle Unit: Choose whether the angle you entered is in "Degrees (°)" or "Radians (rad)" from the dropdown menu.
4. View Results: The calculator will automatically update and display the "Radius (r)" in the results section, along with the angle converted to radians (if you entered degrees) and the value of π used. The formula applied is also shown.
5. Reset or Copy: You can click "Reset" to clear the fields to their default values or "Copy Results" to copy the main result and intermediate values to your clipboard.

The dynamic table and chart also update based on your inputs, providing more visual context.

Key Factors That Affect Radius of Circle with Arc Length Results

The calculated radius depends directly on a few key factors:

1. Arc Length (L): The radius is directly proportional to the arc length for a fixed angle. If you increase the arc length while keeping the angle constant, the radius of the circle must be larger.
2. Angle (θ): The radius is inversely proportional to the angle for a fixed arc length. If you increase the angle while keeping the arc length constant, the radius must be smaller to "fit" the same arc length into a larger angle.
3. Angle Unit: The formula used by the radius of circle with arc length calculator changes depending on whether the angle is in degrees or radians. It's crucial to select the correct unit.
4. Precision of π: The value of π used in the calculation (if the angle is in degrees) affects the precision of the result. Our calculator uses a high-precision value of π.
5. Measurement Accuracy: The accuracy of the calculated radius is directly dependent on the accuracy of the input arc length and angle measurements.
6. Positive Values: Arc length and angle (for practical purposes here) should be positive values to yield a meaningful positive radius.

Understanding these factors helps in interpreting the results from the radius of circle with arc length calculator.

Frequently Asked Questions (FAQ)

1. What is an arc length?

An arc length is the distance along the curved line making up the arc, which is a segment of the circumference of a circle.

2. What is a central angle?

A central angle is an angle whose vertex (corner) is the center of the circle and whose sides pass through the endpoints of an arc or chord.

3. Why are radians used in the simpler formula?

Radians are a "natural" unit for measuring angles in mathematics, defined such that an angle of 1 radian subtends an arc of length equal to the radius. This definition leads to the simple formula L = r * θ.

4. Can I use the radius of circle with arc length calculator for any angle?

Yes, as long as the angle is positive. Typically, angles are between 0 and 360 degrees (or 0 and 2π radians), but the formula works mathematically for larger angles too, representing multiple rotations.

5. What if my arc length and angle give a very small or large radius?

The radius of circle with arc length calculator will give you the mathematically correct radius based on your inputs. If the angle is very small for a given arc length, the radius will be very large, and vice-versa.

6. How accurate is this radius of circle with arc length calculator?

The calculator uses standard mathematical formulas and a high-precision value of π, so its accuracy is primarily limited by the precision of your input values.

7. What are the units of the radius?

The units of the calculated radius will be the same as the units you used for the arc length.

8. Can I calculate arc length or angle if I know the radius?

Yes, by rearranging the formulas: L = r * θ (θ in radians) or L = r * θ * (π / 180) (θ in degrees), and θ = L / r (for radians) or θ = (L * 180) / (r * π) (for degrees). You might need a different calculator for those.

Related Tools and Internal Resources