Curve Angle Calculator: Find the Angle of a Curve

A curve angle calculator finds the change in direction along a circular curve from its radius and arc length. Enter the radius and the distance measured along the curve; the calculator returns the central angle in degrees, along with its equivalent in radians.

Formula

For a circular arc, the angle in radians equals arc length divided by radius. To convert that angle to degrees, multiply by 180 and divide by π:

θ = s / r (radians); θ° = (s / r) × 180 / π (degrees).

• θ is the central angle: the angle at the circle’s center, in radians or degrees.
• s is the arc length, measured along the curved path, in metres.
• r is the circle’s radius, measured from its center to the curve, in metres.

The length and radius must use the same units so their ratio is unitless. The calculator uses metres for both inputs, but you can use any consistent unit in the formula. For example, feet with feet or inches with inches work just as well.

Worked Example

With the default inputs, the radius is 10 m and the arc length is 5 m.

1. Find the angle in radians: θ = s / r = 5 m / 10 m = 0.5 rad.
2. Convert radians to degrees: θ° = 0.5 × 180 / π ≈ 28.6479°.
3. Rounded to the calculator’s two decimal places, the central angle is 28.65°. The equivalent angle is 0.5000 rad.

What a Curve Angle Means

A circular curve has a constant radius, and its angle describes how far the direction turns around the circle’s center. An arc spanning a full circle measures 360 degrees, or 2π radians. A straight segment has no turn; for a circle with a fixed radius, its arc length approaches zero as its central angle approaches zero.

“Curve angle” can mean different things in different fields. This calculation gives the central angle subtended by a circular arc. It does not directly calculate the angle between two lines, the slope of a road, or the changing direction of a non-circular curve. In road and railway design, terminology and conventions can vary, so check what a drawing or specification means by its angle before applying this result.

How to Measure the Inputs

Measure arc length along the curve, not the straight-line distance between the arc’s endpoints. That straight line is called the chord. For a curved path, the chord is shorter than the arc, so substituting chord length for arc length gives an angle that is too small.

The radius must correspond to the same circular curve whose arc length you measured. If a diagram gives diameter rather than radius, divide the diameter by two first. For example, a circle 20 ft across has a radius of 10 ft. Then use arc length and radius in the same unit, such as feet and feet or metres and metres.

When to Use This Calculation

This relationship is useful when a circular arc’s length and radius are known, but its sweep angle is not. It can help with basic geometry, interpreting a circular layout, or checking dimensions in a drawing. It also works in reverse: if you know the angle and radius, arc length is s = rθ, with θ in radians.

For instance, a 12 ft radius curve with a 6 ft arc has θ = 6 / 12 = 0.5 rad, or about 28.65°. The same result follows from a 12 m radius and a 6 m arc. The units change, but the ratio—and therefore the angle—does not.

Common Mistakes and Limits

• Mixing units: Do not divide an arc length in feet by a radius in metres. Convert one measurement so both use the same unit first.
• Using chord length: The chord is not the arc. For a short arc the difference may be modest, but it grows for wider turns.
• Confusing radius and diameter: The radius is half the diameter. Using the diameter as the radius makes the calculated angle half as large for the same arc.
• Forgetting radians in arc formulas: The relation s = rθ requires θ in radians. Convert degrees to radians using θ = degrees × π / 180 before using it.
• Expecting a non-circular curve to have one radius: A curve whose curvature changes does not have a single constant radius over its full length. This calculator cannot describe that shape with one radius and one arc length.

Measurements are often rounded, so the displayed angle is an approximation to the precision of the inputs. Extra decimal places in the answer do not make rough measurements more exact. The formula also assumes the curve is a circular arc and that the entered radius is positive.

Frequently Asked Questions

How do you calculate the angle of a curve?
For a circular curve, divide arc length by radius to get the angle in radians: θ = s / r. Multiply by 180 / π to express it in degrees.

Is curve angle the same as central angle?
In this calculator, yes: “curve angle” means the central angle subtended by a circular arc. Other technical contexts may use curve-angle terminology differently, so confirm the definition used in a plan or specification.

Can I use feet or inches instead of metres?
Yes. The formula works with any length unit as long as arc length and radius use the same one. You can use feet with feet, inches with inches, or metres with metres.

Can I calculate the angle from a chord and radius?
Yes, but that is a different input relationship. For a chord length c and radius r, the minor central angle is θ = 2 asin(c / (2r)) in radians, provided c is no greater than 2r. Do not enter the chord as though it were arc length.

What if the curve is not a perfect circle?
A single radius does not fully describe a curve whose curvature changes. You may need a sequence of local curve measurements or a method designed for the particular path and its geometry.

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in r | Curve radius | m | 10 | 0
in s | Arc length | m | 5 | 0
out angle | Central angle | degrees | (s / r) * 180 / pi | 2
out radians | Central angle | rad | s / r | 4