Angle Converter

How to Calculate Angle Conversions

Angle conversion is a fundamental concept in mathematics and physics, allowing us to express angular measurements in different units. This process involves converting between degrees, radians, and gradians, which are the most common units for measuring angles.

Angle Conversion Formulas

The formulas for converting between different angle units are:

• Degrees to Radians: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\)
• Radians to Degrees: \(\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}\)
• Degrees to Gradians: \(\theta_{grad} = \theta_{deg} \times \frac{10}{9}\)
• Gradians to Degrees: \(\theta_{deg} = \theta_{grad} \times \frac{9}{10}\)

Where:

• \(\theta_{rad}\) is the angle in radians
• \(\theta_{deg}\) is the angle in degrees
• \(\theta_{grad}\) is the angle in gradians
• \(\pi\) (pi) is approximately 3.14159

Calculation Steps

1. Identify the initial angle unit and the desired unit for conversion
2. Select the appropriate formula based on the conversion needed
3. Substitute the given angle value into the formula
4. Perform the calculation to obtain the converted angle

Example Calculation

Let's convert 45 degrees to radians:

1. Identify the conversion: Degrees to Radians
2. Select the formula: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\)
3. Substitute the value: \[\theta_{rad} = 45° \times \frac{\pi}{180}\]
4. Calculate: \[\theta_{rad} = 45 \times 0.01745 = 0.7854 \text{ radians}\]

Therefore, 45 degrees is equal to approximately 0.7854 radians.

Angle Conversion Visualization

This pie chart visually represents a 45-degree angle (shown in blue) within a full 360-degree circle. The diagram helps to illustrate the proportion of the full circle that 45 degrees (or approximately 0.7854 radians) occupies.