RAC Distance Calculator (Radius, Arc, Chord)
Calculate curved distance (arc) and straight-line distance (chord) from a circle radius and central angle.
What is a RAC distance calculator?
A RAC distance calculator is a simple geometry tool that works with three linked ideas: Radius, Arc, and Chord. If you know the radius of a curve and the central angle, you can quickly estimate:
- the arc distance (distance along the curve), and
- the chord distance (straight-line shortcut between endpoints).
This is useful for planning curved tracks, road bends, circular paths, park designs, and any layout where you need both “around the curve” and “as the crow flies” distance.
How the calculator works
The tool uses standard circle formulas. You enter a radius and a central angle in degrees, and the calculator converts the angle to radians behind the scenes.
Core formulas
- Angle in radians: θ = angle × (π / 180)
- Arc length: s = r × θ
- Chord length: c = 2r × sin(θ / 2)
It also shows the difference between arc and chord. That difference grows as the angle gets larger. For very small angles, arc and chord are almost the same.
Why arc and chord distances both matter
In real projects, people often need both numbers at once. The arc length tells you the actual travel or material length on a curve. The chord length helps with direct alignment, line-of-sight estimates, and simplified construction measurements.
- Civil and road design: estimate curved segment lengths.
- Landscape architecture: size curved walkways or edging.
- Sports planning: mark curved sections of tracks and fields.
- DIY projects: build curved rails, fences, and decorative arcs.
Step-by-step usage guide
1) Enter radius
Radius is the distance from the circle center to its edge. Use any unit you want (meters, feet, miles, etc.). The output stays in the same unit.
2) Enter central angle
Enter the angle in degrees between 0 and 360. A larger angle means a longer arc around the circle.
3) Add unit label (optional)
Type a label like m, km, ft, or mi so results are easier to read.
4) Click Calculate Distance
You’ll see arc distance, chord distance, and the extra distance traveled by following the curve instead of a straight line.
Worked example
Suppose your radius is 100 m and your central angle is 90°.
- θ = 90 × (π/180) = π/2
- Arc = 100 × (π/2) ≈ 157.08 m
- Chord = 2 × 100 × sin(45°) ≈ 141.42 m
So the curved path is about 15.66 m longer than the straight-line connection.
Common mistakes to avoid
- Mixing units: keep your radius in one unit system.
- Forgetting angle limits: this page expects degrees from 0 to 360.
- Confusing diameter and radius: radius is half the diameter.
- Using negative values: distances should be positive values.
RAC calculator FAQ
Can I use miles or kilometers?
Yes. Use any distance unit for radius. Results will be in that same unit.
Why is chord always shorter than arc?
A straight line is the shortest path between two points. The arc follows the curve, so it is equal or longer.
What happens at 360°?
A full turn returns to the same point. Arc equals the full circumference, while chord approaches zero.
Final thoughts
A solid rac distance calculator helps you move from rough guesswork to precise planning. Whether you’re designing infrastructure, planning a curved route, or building at home, these radius-arc-chord calculations give you reliable numbers fast.