Geometry plays a critical role in mathematics, physics, construction, and design. Among the most useful concepts is the arc of a circle. An arc is simply a portion of a circle’s circumference, and calculating its area is important in many fields, from architectural designs to physics problems.
Arc Area Calculator
🔎 What is Arc Area?
The arc area refers to the portion of a circle’s area enclosed by an arc and the radii connecting the arc’s endpoints to the center. Depending on the shape, this can either be:
- Sector Area – The area of the “pizza slice” shape formed by two radii and the arc.
- Segment Area – The area between the arc and the chord (a straight line joining the arc’s endpoints).
The Arc Area Calculator focuses on sector area, but it can also help with segment area if the chord is considered.
🛠 How to Use the Arc Area Calculator
Using the tool is simple:
- Input the circle’s radius – Enter the radius (distance from the center to the edge).
- Enter the arc angle or arc length – Depending on which values you have.
- Click calculate – The tool will instantly compute the arc’s area.
- View results – You’ll see the arc area along with related values like sector perimeter.
This saves time and ensures accuracy, especially for engineering or academic use.
📐 Formula for Arc Area
There are two main formulas depending on what information you have:
1. Arc Area (using angle θ in degrees)
Arc Area = (θ ÷ 360) × π × r²
Where:
- θ = central angle in degrees
- r = radius of the circle
2. Arc Area (using angle θ in radians)
Arc Area = (1/2) × r² × θ
Where:
- θ = angle in radians
- r = radius
3. Arc Area (using arc length L)
Arc Area = (L × r) ÷ 2
Where:
- L = arc length
- r = radius
These formulas ensure flexibility no matter what values you know.
✅ Example Calculations
Example 1 – Using Degrees
Find the arc area of a circle with radius 10 cm and angle 60°.
Arc Area = (60 ÷ 360) × π × 10²
Arc Area = (1/6) × 3.1416 × 100
Arc Area ≈ 52.36 cm²
Example 2 – Using Radians
Find the arc area with radius 8 cm and angle 1.5 radians.
Arc Area = (1/2) × 8² × 1.5
Arc Area = 48 cm²
Example 3 – Using Arc Length
Find arc area if arc length = 12 cm and radius = 6 cm.
Arc Area = (12 × 6) ÷ 2
Arc Area = 36 cm²
🎯 Practical Applications
The Arc Area Calculator is useful in:
- Architecture & Engineering – Designing curved structures and windows.
- Construction – Calculating material for circular sections like arches.
- Mathematics & Geometry – Solving problems in exams and academic research.
- Physics – Measuring rotational motion and circular paths.
- Design & Art – Creating arcs in patterns and structures.
⚖️ Arc Area vs. Circle Area
- Circle Area Formula = π × r² (entire circle).
- Arc Area Formula = (θ ÷ 360) × π × r² (just part of the circle).
This shows that an arc area is simply a fraction of the total circle area.
💡 Tips for Accurate Calculations
- Convert angles to radians when using advanced formulas.
- Always use the same units for radius and arc length.
- Use π ≈ 3.1416 for quick estimates or leave as π for exact results.
- For segment area, subtract triangle area from the sector area.
❓ 20 Frequently Asked Questions (FAQs)
Q1. What is an arc?
An arc is a portion of the circumference of a circle.
Q2. What is arc area?
It is the area covered by a sector formed by an arc and two radii.
Q3. How do I calculate arc area from angle in degrees?
Use formula: (θ ÷ 360) × π × r².
Q4. What if the angle is in radians?
Use formula: (1/2) × r² × θ.
Q5. Can I calculate arc area using arc length?
Yes, use formula: (L × r) ÷ 2.
Q6. What’s the difference between arc length and arc area?
Arc length measures distance along the arc, while arc area measures the space inside it.
Q7. How is sector area different from segment area?
Sector is a “slice” of the circle; segment is formed by arc and chord.
Q8. Can arc area be bigger than half the circle?
Yes, if the angle is greater than 180°.
Q9. What units are used for arc area?
Square units (cm², m², etc.).
Q10. Do I need π value for calculations?
Yes, π is always used in circle-based formulas.
Q11. Can I calculate arc area without radius?
No, radius is essential.
Q12. Is arc area used in real life?
Yes, in construction, design, and geometry.
Q13. How do I convert degrees to radians?
Radians = (π ÷ 180) × Degrees.
Q14. Can I use arc area formula for ellipses?
No, these formulas only apply to circles.
Q15. What is a central angle?
The angle formed at the circle’s center by two radii.
Q16. Can arc area be calculated with diameter?
Yes, since radius = diameter ÷ 2.
Q17. How to find arc length from arc area?
Use L = (2 × Arc Area) ÷ r.
Q18. Is arc area always less than circle area?
Yes, except when it’s the full circle.
Q19. Do calculators give exact or approximate values?
Most give approximate decimal values.
Q20. Why use an online Arc Area Calculator?
It saves time, avoids mistakes, and handles complex inputs instantly.
🏁 Final Thoughts
The Arc Area Calculator is an essential tool for anyone working with circles. It simplifies the process of finding the area of circular arcs using radius, angle, or arc length. From construction projects to academic studies, this tool ensures quick, accurate, and reliable results.