Arc Length Calculator - Mathematical Calculations & Solutions
How It Works
Enter Arc Data
Radius and angle
Calculate Arc Length
Apply s = rθ
Common Examples
What is an Arc Length Calculator?
An arc length calculator is a simple tool that helps you find the length of a curved line on a circle. Think of it like measuring how long a piece of string would be if you laid it along the edge of a circle.
This calculator is very useful when you need to measure curved distances. It helps engineers design roads, architects plan buildings, and students solve math problems.
Easy Examples
- •Measuring the length of a curved road
- •Finding how far you walk on a circular track
- •Calculating the length of a pizza slice edge
- •Measuring the curve of a bridge or arch
Why It Helps
- •Design curved roads and highways safely
- •Build bridges and arches with correct measurements
- •Solve geometry problems in school and work
- •Calculate materials needed for curved projects
How to Use This Arc Length Calculator - Simple Steps
Step 1: Understand What You Need
Radius: Distance from center to edge of circle
Example: If a circle is 10 feet across, the radius is 5 feet
Central Angle: How much of the circle you want
Example: 90 degrees is a quarter of the circle
Arc Length: The curved distance you get
Example: How long the curved line is
Step 2: Enter Your Numbers
Choose Angle Unit: Degrees or Radians
Use degrees for everyday problems (easier to understand)
Enter Radius: Distance from center to edge
Use any unit like feet, meters, inches, etc.
Enter Angle: How much of the circle
Like 90° for quarter circle, 180° for half circle
Get Result: Arc length appears automatically
Shows the curved distance in same unit as radius
Quick Example to Try
Try these numbers to see how it works:
Radius: 5 (like 5 feet)
Angle: 90 degrees
Unit: Degrees
Arc Length: About 7.85 feet
This is the length of a quarter circle with 5-foot radius
Real World Examples
🛣️ Road Design
Engineers use arc length to design curved roads and highways. They need to know exactly how long each curve is for safety and construction.
Example:
Highway curve: Radius 500m, Angle 45°
Curve Length: About 393 meters
🏗️ Bridge Construction
Architects calculate arc lengths when designing curved bridges and arches. This helps them order the right amount of materials.
Example:
Bridge arch: Radius 20m, Angle 120°
Arch Length: About 41.9 meters
🏃♂️ Running Tracks
Sports facilities use arc length calculations to design running tracks. They need to make sure each lane is exactly the right distance.
Example:
Track curve: Radius 36.5m, Angle 180°
Curve Length: About 114.7 meters
⚙️ Mechanical Parts
Engineers design gears, pulleys, and other round mechanical parts. Arc length helps them calculate belt lengths and gear sizes.
Example:
Gear section: Radius 8cm, Angle 60°
Arc Length: About 8.38 cm
🌙 Astronomy
Astronomers use arc length to measure distances in space and calculate orbital paths of planets and satellites.
Example:
Satellite orbit: Radius 7000km, Angle 30°
Arc Distance: About 3665 km
🎡 Amusement Parks
Designers of ferris wheels and roller coasters use arc length to plan the ride experience and safety features.
Example:
Ferris wheel: Radius 25m, Angle 90°
Quarter Ride: About 39.3 meters
The Math Made Simple
What We Calculate
The arc length calculator uses a simple formula to find curved distances. Don't worry - the calculator does all the hard work for you!
The Formula: s = r × θ
- • s = arc length (what we want to find)
- • r = radius (distance from center to edge)
- • θ = angle (in radians, not degrees)
Converting Degrees to Radians
To use the formula, we need to convert degrees to radians. We multiply degrees by π/180. The calculator does this automatically!
Simple Example
Let's find the arc length of a pizza slice:
Given:
Pizza radius = 8 inches
Slice angle = 45 degrees
Step 1: Convert degrees to radians
45° × π/180 = 0.785 radians
Step 2: Apply formula
s = 8 × 0.785 = 6.28 inches
Answer:
The curved edge of the pizza slice is 6.28 inches long
Understanding Circles and Arcs
Circle Parts
Radius
The distance from the center of the circle to any point on the edge. Like the spoke of a bicycle wheel.
Example: If a circle is 10 feet across, the radius is 5 feet.
Diameter
The distance across the whole circle through the center. Always twice the radius.
Example: If radius is 5 feet, diameter is 10 feet.
Circumference
The distance all the way around the circle. Like the perimeter of a square, but curved.
Formula: C = 2πr (about 6.28 times the radius)
Angles and Arcs
Central Angle
The angle at the center of the circle. Like opening a pair of scissors.
Common angles: 90° = quarter circle, 180° = half circle, 360° = full circle
Arc
A piece of the circle's edge. Like a slice of the circumference.
Example: A quarter of a pizza's crust is an arc.
Arc Length
How long the arc is if you measured it with a string.
Key point: Bigger angles = longer arcs, bigger radius = longer arcs
Common Questions People Ask
Basic Questions
What if I only know the diameter?
Just divide the diameter by 2 to get the radius. For example, if diameter is 20 feet, radius is 10 feet.
Should I use degrees or radians?
Use degrees for everyday problems - they're easier to understand. The calculator converts to radians automatically.
What if my angle is bigger than 360°?
That means you're going around the circle more than once. The calculator will still work correctly.
Practical Questions
How accurate are the results?
Very accurate! The calculator uses precise math formulas that engineers and architects use for real projects.
What units can I use?
Any unit works - inches, feet, meters, miles, etc. Just make sure to use the same unit for radius and arc length.
Can I find the angle if I know arc length?
Yes! Rearrange the formula: angle = arc length ÷ radius. This gives you the angle in radians.
Practice Problems to Try
Pizza Slice
Radius: 6 inches
Angle: 60 degrees
Try it! Arc Length: About 6.28 inches
Garden Path
Radius: 10 feet
Angle: 90 degrees
Try it! Arc Length: About 15.71 feet
Race Track Curve
Radius: 50 meters
Angle: 45 degrees
Try it! Arc Length: About 39.27 meters
Clock Hand
Radius: 4 inches
Angle: 30 degrees (1 hour)
Try it! Arc Length: About 2.09 inches
Bridge Arch
Radius: 15 meters
Angle: 120 degrees
Try it! Arc Length: About 31.42 meters
Ferris Wheel
Radius: 20 meters
Angle: 180 degrees (half)
Try it! Arc Length: About 62.83 meters
How to Practice
- Pick a problem from above
- Enter the radius and angle in the calculator
- Check if your answer matches the expected result
- Try to visualize what the arc would look like in real life
- Experiment with different numbers to see how results change
- Try converting between degrees and radians
Arc Length Calculation Table
| Radius | Angle (Degrees) | Angle (Radians) | Arc Length | Common Use |
|---|---|---|---|---|
| 5 | 90° | π/2 | 7.85 | Quarter circle |
| 10 | 60° | π/3 | 10.47 | Road curve |
| 8 | 45° | π/4 | 6.28 | Pizza slice |
| 15 | 120° | 2π/3 | 31.42 | Bridge arch |
| 20 | 180° | π | 62.83 | Semicircle |
| 12 | 30° | π/6 | 6.28 | Clock hour |
*All calculations use the formula s = r × θ (where θ is in radians)
Arc Length Calculator
What
Calculate the length of an arc along the circumference of a circle.
Why
Essential for geometry, engineering, architecture, and circular motion analysis.
Applications
Road design, mechanical engineering, astronomy, and construction projects.
Calculation Examples
| Input | Formula | Result | Use Case |
|---|---|---|---|
| r = 5, θ = 60° | s = 5 × (60° × π/180) | 5.24 | Circle sector |
| r = 10, θ = π/2 rad | s = 10 × π/2 | 15.71 | Quarter circle |
| r = 3, θ = 120° | s = 3 × (120° × π/180) | 6.28 | Arc measurement |
| r = 8, θ = π rad | s = 8 × π | 25.13 | Semicircle |
Frequently Asked Questions
What is arc length?
Arc length is the distance along the curved line of a circle's circumference between two points, measured by the central angle.
What is the arc length formula?
The formula is s = r × θ, where s is arc length, r is radius, and θ is the central angle in radians.
How do I convert degrees to radians?
To convert degrees to radians, multiply by π/180. For example: 60° = 60 × π/180 = π/3 radians.
What are common applications of arc length?
Used in road design, bridge construction, gear design, astronomy, and any circular motion calculations.
Can I calculate arc length with diameter instead of radius?
Yes, use s = (d/2) × θ where d is diameter, or simply divide the diameter by 2 to get the radius first.
What if I have the arc length and need to find the angle?
Rearrange the formula to θ = s/r. This gives you the central angle in radians when you know arc length and radius.
How accurate are the calculations?
The calculator uses precise mathematical formulas with high precision, suitable for engineering and scientific applications.