Question

Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. \(18.255^{\circ}\)

Short answer

18° 15′ 18″

Step-by-step solution

- Separate the Degrees

The given angle is in decimal degrees. First, separate the whole number part, which is the degrees. In this case, it's 18 degrees.

- Multiply the Decimal by 60

Take the decimal part (0.255) and multiply it by 60 to convert it into minutes. 0.255 × 60 = 15.3

- Separate the Minutes

The whole number from the result in Step 2 is the minutes. Here, it is 15 minutes.

- Multiply the Decimal Minutes by 60

Now take the decimal part of the minutes (0.3) and multiply by 60 to convert it into seconds. 0.3 × 60 = 18

- Round the Seconds

Round the seconds to the nearest whole number if necessary. In this case, 18 is already a whole number.

- Combine the Results

Combine the separated degrees, minutes, and seconds. Therefore, 18.255 degrees can be written as 18° 15′ 18″.

Key concepts

Decimal Degrees

When working with angles, you might often see them expressed in decimal degrees. Decimal degrees are a way to represent angles using a decimal point instead of separating degrees, minutes, and seconds. Instead of saying an angle measures 18 degrees, 15 minutes, and 18 seconds, you can say it measures 18.255 degrees.
Decimal degrees make calculations easier, especially when using a calculator or computer. This is because you can perform basic arithmetic operations without converting back and forth between degrees, minutes, and seconds.
To convert an angle from decimal degrees to degrees, minutes, and seconds, follow these steps:

• Separate the whole number part as the degrees.
• Take the decimal part and multiply it by 60 to get the minutes.
• If the result is not a whole number, take the decimal part of the minutes and multiply it by 60 to get the seconds.

Degrees Minutes Seconds

Angles can also be expressed in a more traditional way using degrees, minutes, and seconds (D° M′ S″). Degrees are the largest unit, and there are 60 minutes in one degree and 60 seconds in one minute.
To convert an angle from decimal degrees to this format, you start by separating out the whole number part of the decimal as the degrees. Next, multiply the remaining decimal portion by 60 to get the minutes. If there is a decimal part in the minutes, that indicates the seconds, and you need to multiply that decimal by 60 to find the seconds.

• For example, 18.255 degrees is converted as follows:
• 18 is the degrees.
• 0.255 multiplied by 60 equals 15.3, so 15 is the minutes.
• 0.3 multiplied by 60 equals 18, hence the seconds are 18.

This results in the angle being written as 18° 15′ 18″.

Rounding

Rounding plays an important role when converting angles into degrees, minutes, and seconds. This process helps to avoid long decimal fractions and makes the angle easier to read and use.
Here are a few guidelines:

• When you have a decimal number of seconds, you round it to the nearest whole number.
• If the seconds decimal is exactly 0.5 or higher, round up. Otherwise, round down.

In our example, 18.3 seconds would round down to 18 because it is less than 18.5. If it was 18.5 or more, it rounds up to 19. Understanding rounding ensures that your angles are accurate to the required precision.
To sum up, rounding simplifies your final result for ease of reading and application.