Arc Calculator

An arc is a segment of a circle’s circumference, defined by two points on its edge. Calculating the arc length or the angle it spans is an essential task in geometry, construction, engineering, navigation, and even everyday problem-solving. The Arc Calculator is a powerful tool that allows you to find arc length, radius, central angle, or other related values quickly and accurately.

Arc Calculator

How to Use the Arc Calculator

Using the Arc Calculator is straightforward. Just follow these steps:

  1. Select the quantity you want to find – You can calculate arc length, central angle, or radius.
  2. Enter the known values – Depending on what you are solving for, you will need two of the following:
    • Radius
    • Arc length
    • Central angle (in degrees or radians)
  3. Click "Calculate" – The tool will instantly compute the result.

Example Inputs:

  • Arc Length Mode: Enter the radius and central angle to find arc length.
  • Angle Mode: Enter arc length and radius to find the angle.
  • Radius Mode: Enter arc length and angle to find the radius.

Formulas Used in the Arc Calculator

The Arc Calculator is based on standard circle geometry formulas.

1. Arc Length Formula

Arc Length (L) = (θ / 360) × 2πr
Where:

  • L = Arc length
  • θ = Central angle in degrees
  • r = Radius of the circle

2. Arc Length Formula (Radians)

Arc Length (L) = r × θ
Where:

  • θ is in radians

3. Central Angle Formula

Central Angle (Degrees) = (L / (2πr)) × 360

4. Radius Formula

Radius (r) = L × 360 / (2π × θ)

Example Calculations

Example 1 – Finding Arc Length

You have a circle with a radius of 10 m and a central angle of 60°.
L = (60 / 360) × 2π × 10
L = (1/6) × 20π ≈ 10.472 m

Result: The arc length is about 10.47 meters.

Example 2 – Finding Central Angle

You have an arc length of 15 m and a radius of 12 m.
Central Angle = (15 / (2π × 12)) × 360
= (15 / 75.398) × 360 ≈ 71.64°

Result: The central angle is approximately 71.64 degrees.

Example 3 – Finding Radius

You have an arc length of 20 m and a central angle of 90°.
Radius = (20 × 360) / (2π × 90)
= 7200 / 565.486 ≈ 12.73 m

Result: The radius is about 12.73 meters.

Why Use an Arc Calculator?

  • Accuracy – No manual miscalculations.
  • Speed – Instant results for multiple inputs.
  • Versatility – Works with degrees or radians.
  • Educational Value – Helps students understand geometry concepts better.
  • Practical Applications – Useful in architecture, landscaping, road design, and mechanical parts manufacturing.

Real-World Applications

  1. Construction – Calculating curved railings or archways.
  2. Engineering – Designing gears, wheels, and circular components.
  3. Road Design – Measuring bends and curves.
  4. Art & Design – Creating circular patterns or arcs.
  5. Navigation – Measuring distances along a curved path.

Tips for Accurate Arc Calculations

  • Always ensure angles are in the correct unit (degrees or radians).
  • Use consistent units for radius and arc length.
  • Double-check your inputs before calculating.
  • For very large circles, remember small rounding differences can affect results.

20 Frequently Asked Questions (FAQs)

1. What is an arc?
An arc is a portion of the circumference of a circle.

2. How do I calculate arc length?
Use the formula L = (θ / 360) × 2πr for degrees or L = r × θ for radians.

3. Can the calculator work with radians?
Yes, it can handle both degrees and radians.

4. What inputs are needed to find arc length?
You need the radius and central angle.

5. What inputs are needed to find the central angle?
You need the arc length and radius.

6. What inputs are needed to find the radius?
You need the arc length and central angle.

7. Can this calculator be used for semicircles?
Yes, simply enter 180° as the central angle.

8. Does this work for ellipses?
No, it’s designed for perfect circles.

9. Is the formula accurate for large circles?
Yes, it’s accurate for any circle size.

10. Can I use the calculator for irregular curves?
No, only for perfect circular arcs.

11. Is the arc length the same as chord length?
No, the chord is a straight line between two points, while arc length follows the curve.

12. Can I calculate chord length from arc length?
Yes, but it requires an additional formula.

13. Can this be used in construction planning?
Absolutely, it’s very useful for curved designs.

14. How do I switch between degrees and radians?
Use a conversion: radians = degrees × (π / 180).

15. Does the calculator handle negative angles?
No, angles should be positive.

16. Is this tool suitable for students?
Yes, it’s great for learning geometry.

17. Can I use different units?
Yes, as long as they’re consistent.

18. Is there a mobile-friendly version?
Yes, most arc calculators work on mobile devices.

19. Does the calculator save my data?
No, it processes inputs instantly without storing them.

20. Is this tool free?
Yes, it’s completely free to use online.