Degrees to Meters Calculator: Convert Angles to Distance

This calculator converts an angle in degrees and a radius (or distance from the center) into the arc length in meters. You enter the angle in degrees and the radius in meters, and you get back the arc length in meters, plus the equivalent in centimeters and kilometers. This is useful for surveying, map work, circular motion, and any geometry problem where you need to know how far a point moves along a circle.

Formula

The arc length \( L \) (in meters) for an angle \( \theta \) (in degrees) and radius \( r \) (in meters) is:

\( L = \frac{\theta}{360} \times 2\pi r \)

Where:

• \( L \) = arc length in meters (the distance along the circle)
• \( \theta \) = central angle in degrees (0 to 360 for a full circle)
• \( r \) = radius in meters (distance from the center to the arc)
• \( \pi \approx 3.14159 \)

The factor \( \frac{\theta}{360} \) is the fraction of the full circle that the angle represents. Multiply that by the circumference \( 2\pi r \) to get the arc length.

Worked Example

Using the default values: angle = 45°, radius = 10 m.

1. Find the circumference: \( 2 \times \pi \times 10 = 62.8319 \) m
2. Find the fraction of the circle: \( 45 / 360 = 0.125 \)
3. Multiply: \( 0.125 \times 62.8319 = 7.854 \) m

The calculator shows 7.85 m (rounded to 2 decimals). In centimeters that is 785 cm, and in kilometers 0.0079 km.

When to Use This Conversion

You need this conversion whenever you have an angle and a radius and want a linear distance. Common situations:

• Surveying: measuring distances along a curved boundary or road.
• Map reading: converting a bearing change and distance from a point into a ground distance along a circle.
• Machining and design: finding how far a point on a rotating part travels.
• Physics: calculating linear displacement from angular displacement (e.g., a wheel rolling).
• Sports and fitness: distance run on a circular track given the angle turned.

The result is the distance along the arc, not the straight-line (chord) distance. If you need the straight line between two points on a circle, use the chord formula instead: \( 2r \sin(\theta/2) \).

How to Choose the Radius

The radius is the distance from the center of the circle to the point where you want the arc length. For example:

• If you are standing at the center and walk 10 m out, then turn 45°, your arc length is based on radius = 10 m.
• If you are measuring along the Earth’s surface, the radius is the Earth’s radius (about 6,371,000 m) plus your altitude.
• If you are measuring a curve on a road, the radius is the road’s curvature radius (often given in design plans).

Make sure the radius and the result are in the same unit. If your radius is in feet, convert to meters first (1 ft = 0.3048 m) or use a calculator that accepts feet.

Common Mistakes and Misconceptions

A frequent error is mixing degrees with radians. This calculator uses degrees, but many formulas in physics and engineering use radians. To convert degrees to radians, multiply by \( \pi/180 \). If you accidentally use radians in this calculator, your result will be wrong by a factor of \( 180/\pi \approx 57.3 \).

Another mistake is confusing arc length with chord length. For a 45° angle on a 10 m radius, the arc is 7.85 m, but the straight-line distance between the two endpoints is only \( 2 \times 10 \times \sin(22.5°) = 7.65 \) m. The difference is small for small angles but grows as the angle increases.

Also, remember that the angle must be between 0 and 360 degrees for a single arc. If you have an angle larger than 360°, it means more than one full revolution, and the arc length would be the remainder plus full circumferences. This calculator assumes a single angle ≤ 360°.

Limits of the Formula

The formula assumes a perfect circle and a constant radius. In real-world applications, the Earth is not a perfect sphere (it is slightly oblate), so surveying over large distances requires ellipsoidal models. For small areas (less than a few kilometers), the error is negligible.

The formula also gives the arc length along the circle, not the actual ground distance if the terrain is hilly. For hilly terrain, you need to account for elevation changes separately.

For very small angles (less than about 1°), the arc length is nearly equal to the chord length, and many practical measurements use the chord approximation. But for precision, use the exact formula.

Alternatives and Related Calculations

If you need to convert degrees to meters directly without a radius, that is impossible because an angle has no inherent length. You must know the radius. This is a common misconception: people ask “how many meters in a degree?â€� but the answer depends on the circle size. On the Earth’s surface, one degree of latitude is about 111,320 m, but one degree of longitude varies from 111,320 m at the equator to 0 m at the poles.

Related calculations you might need:

• Chord length: straight-line distance between two points on a circle.
• Sector area: area enclosed by the arc and two radii.
• Angular velocity to linear velocity: \( v = \omega r \), where \( \omega \) is in radians per second.
• Degrees to radians: multiply by \( \pi/180 \).

Frequently Asked Questions

How many meters are in one degree?

It depends on the radius. On a circle of radius \( r \), one degree of arc is \( \frac{2\pi r}{360} \). For Earth’s equator (radius ≈ 6,378 km), one degree of longitude is about 111.32 km. But for a small circle (like a roundabout with radius 10 m), one degree is only about 0.175 m.

Can I convert degrees to meters without a radius?

No. An angle is dimensionless; it only becomes a length when multiplied by a radius. You must know the distance from the center of rotation to the point of interest.

What is the difference between arc length and chord length?

Arc length is the distance along the curved path. Chord length is the straight-line distance between the two endpoints. For a given angle and radius, chord length is always shorter than arc length (except for 0°).

Does this work for angles in radians?

No, this calculator expects degrees. If you have radians, convert to degrees first by multiplying by \( 180/\pi \), or use the formula \( L = r \theta \) with \( \theta \) in radians.

Why does the result have so many decimal places?

The calculator shows 2 decimals for meters, which is usually precise enough for most practical purposes. For very large radii (like Earth), you might want more precision, so the kilometer output shows 4 decimals.

Is this the same as converting latitude/longitude to meters?

Not directly. Latitude and longitude are angles on a sphere, and converting them to meters requires a model of the Earth (e.g., WGS84). The arc length formula applies to a flat circle. For small areas, you can approximate, but for global distances, use a geodesic calculator.

Related

• Area Calculator Circle Diameter: Formula & Examples
• Area Calculator Converter: How to Convert Units
• Arc Length to Diameter Calculator

in angle | Angle | degrees | 45 | 0
in radius | Radius | m | 10 | 0
out arc | Arc length | m | (angle / 360) * 2 * pi * radius | 2
out arc_cm | Arc length | cm | arc * 100 | 0
out arc_km | Arc length | km | arc / 1000 | 4