Angle unit converter

Angles are an integral part of geometry, trigonometry, and many other scientific and engineering disciplines. Various units are used to describe them, and it is often necessary to transform one format into another. The angle unit converter can instantly convert angle values from one unit to another, ensuring high precision and convenience. In this article, we will explore the main angle measurement units, their applications, and transformation methods.

What are angle measurement units?

Angle measurement units are standardized ways to express the magnitude of angles. Different units can be used depending on the discipline and practical application. The main angle units that our converter operates with are:

Degrees (°): A widely used unit that is intuitive to understand. A full circle is 360 degrees.
Radians (rad): The primary angular measurement unit in mathematics and physics. A full circle is $2\pi$ radians.
Degrees, minutes, seconds (° ’ ”): A more precise expression of an angle, widely used in astronomy and navigation. One degree equals 60 arc minutes, and each minute equals 60 arc seconds.
Gradians (gon): Used in geodesy and some engineering applications, a full circle is 400 gradians.

Arc minutes and arc seconds

Arc minutes and arc seconds divide an angle into smaller parts, allowing for more precise measurements. This is particularly important in fields requiring high precision, such as cartography and astronomy. 1 degree equals 60 arc minutes, and 1 arc minute equals 60 arc seconds. For instance, 32 degrees, 15 minutes, and 20 seconds are written as $32° 15' 20''$ .

Applications

Navigation: Directions and coordinates are precisely represented using angles with arc minutes and seconds.
Trigonometry: Radians are used in trigonometric functions because they offer a more convenient mathematical representation.
Geodesy: Gradians are helpful for working with angular measurements on large-scale projects.

All our calculators, including the triangle area calculator, can automatically convert degrees to radians and vice versa. This is essential in calculations involving angle formulas, such as finding the area of a triangle through an angle.

Conversion formulas

Conversion between different angle measurement units uses the following formulas:

1 \text{°} = \frac{\pi}{180} \text{ rad}

1 \text{ rad} = \frac{180}{\pi} \text{°}

1 \text{ gon} = 0.9 \text{°}

1 \text{°} = \frac{10}{9} \text{ gon}

1° = 60' \text{ arc minutes}

1' = 60'' \text{ arc seconds}

Conversion examples

Let’s consider several conversion examples:

30 degrees to radians:
$30° \times \frac{\pi}{180} = \frac{\pi}{6} \approx 0.5236 \text{ rad}$
1 radian to degrees:
$1 \text{ rad} \times \frac{180}{\pi} \approx 57.2958°$
200 gradians to degrees:
$200 \text{ gon} \times 0.9 = 180°$
15 arc minutes to degrees:
$15' = \frac{15}{60} = 0.25°$
90 arc seconds to degrees:
$90'' = \frac{90}{3600} = 0.025°$

Angle conversion table

Degrees (°)	Radians (rad)	Gradians (gon)	DMS (° ’ ”)
0°	0 rad	0 gon	0° 0’ 0”
30°	0.5236 rad	33.333 gon	30° 0’ 0”
45°	0.7854 rad	50 gon	45° 0’ 0”
60°	1.0472 rad	66.667 gon	60° 0’ 0”
90°	1.5708 rad	100 gon	90° 0’ 0”
180°	3.1416 rad	200 gon	180° 0’ 0”

Notes

When using the angle converter, it’s important to consider that the accuracy of the conversion depends on the rounding level chosen and the context in which calculations are used.

Frequently asked questions

How to convert 2 radians to gradians?

Use the formulas: first convert radians to degrees, then to gradians.

2 \text{ rad} \times \frac{180}{\pi} \approx 114.59°

114.59° \times \frac{10}{9} \approx 127.32 \text{ gon}

How to convert 15 arc minutes to radians?

First convert arc minutes to degrees, then to radians:

15' = \frac{15}{60} = 0.25°

0.25° \times \frac{\pi}{180} \approx 0.00436 \text{ rad}

How to convert 100 arc seconds to degrees and radians?

First convert arc seconds to degrees, then to radians:

100'' = \frac{100}{3600} = 0.02778°

0.02778° \times \frac{\pi}{180} \approx 0.0004856 \text{ rad}

Why are radians often used in trigonometric functions?

Radians provide more correct mathematical properties and forms for trigonometric functions, making them preferable for mathematical calculations.

Where are arc minutes and arc seconds most commonly used?

Arc minutes and arc seconds are commonly used in astronomy and navigation for precise measurements and coordinate determination.

Angle unit converter

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