Question

Convert to degrees (decimal). $$72^{\circ} 12^{\prime} 22^{\prime \prime}$$

Short answer

72.2061111 degrees

Step-by-step solution

Understand the Conversion

Degrees, minutes, and seconds (DMS) can be converted to decimal degrees by using the conversion that 1 minute is equal to 1/60 of a degree, and 1 second is equal to 1/3600 of a degree. The given measure is in DMS, which needs to be converted to decimal degrees.

Convert Minutes to Degrees

Convert the minutes to decimal degrees by dividing the number of minutes by 60. In this case, you have 12 minutes, which is equal to \( 12 \div 60 = 0.2 \) degrees.

Convert Seconds to Degrees

Similarly, convert the seconds to decimal degrees by dividing the number of seconds by 3600. So, 22 seconds equals \( 22 \div 3600 \approx 0.0061111 \) degrees.

Sum All Degrees

Add the decimal degrees obtained from minutes and seconds to the whole degrees to get the final answer in decimal. \( 72^{\text{o}} + 0.2 + 0.0061111 = 72.2061111^{\text{o}} \).

Key concepts

Angle Conversion

Angle conversion is a fundamental concept in geometry and various real-world applications, such as navigation, surveying, and astronomy. Angles can be expressed in different units, with the most common being degrees, minutes, and seconds (DMS) and decimal degrees. To perform accurate measurements, calculations, and data analysis, it's often necessary to convert angles from one unit to another.

When converting angles, it's important to understand the relationship between the units. A degree is divided into 60 minutes, and one minute is further divided into 60 seconds. Converting an angle from DMS to decimal degrees involves translating these subdivisions into a single decimal value that represents the angle more simply. This process improves the calculation's efficiency, especially when inputting data into most modern computational tools and software that prefer decimal degrees.

Degrees Minutes Seconds

Degrees, minutes, and seconds (DMS) is a notation that divides one degree into 60 minutes and one minute into 60 seconds. This form of notation is derived from the historical use of base-60 mathematics, also known as sexagesimal notation. It's particularly useful for precision when smaller angular measurements are needed without resorting to decimals.

For example, when using a compass or reading maps, an angle might be given in DMS to provide clear and precise direction to the user. Understanding how to read and write angles in DMS is crucial for tasks that require high precision. It's also a stepping stone to mastering angle conversions, as it lays the groundwork for understanding the hierarchy of angle units.

Decimal Degrees

Decimal degrees are an alternative representation of angles, where the value is expressed as a simple decimal number. This format is more intuitive for many calculations and is commonly used in fields such as GIS (Geographic Information Systems) and GPS technology, where coordinates are often needed in a continuous decimal format for ease of computation and readability.

The process of converting from DMS to decimal degrees involves consolidating minutes and seconds into a fraction of a degree, where minutes are divided by 60 and seconds by 3600, and then adding those to the whole degrees to obtain a final decimal value. For instance, an angle of 72 degrees, 12 minutes, and 22 seconds is converted to decimal degrees by the conversion formula: sum the whole number of degrees (72), the minutes divided by 60 (12/60 = 0.2), and the seconds divided by 3600 (22/3600 ≈ 0.0061), which gives a result of approximately 72.2061 degrees.