How to Convert Degrees to Minutes and Seconds? Complete Guide

Introduction

Angles are commonly measured in degrees (°), minutes (′), and seconds (″). In mathematics, it is often necessary to convert an angle given in degrees into degrees, minutes and seconds, or to convert minutes and seconds back into degrees.

Understanding these conversions is important for students studying geometry, trigonometry, and other mathematical topics.

In this article, we will learn how to convert degree into minutes and seconds, how to change decimal degrees to degrees, minutes and seconds, and how to convert minutes and seconds to degrees.

What Are Degrees, Minutes and Seconds?

An angle can be divided into smaller units as follows:

• 1 degree = 60 minutes
• 1 minute = 60 seconds
• Therefore, 1 degree = 3600 seconds

The symbols used are:

• Degree = °
• Minute = ′
• Second = ″

For example:

25° 30′ 20″

is read as:

25 degrees, 30 minutes and 20 seconds.

Relationship Between Degrees, Minutes and Seconds

The basic relationships are:

1° = 60′

and

1′ = 60″

Therefore:

1° = 60 × 60″

1° = 3600″

These relationships are the foundation for all conversions between degrees, minutes and seconds.

How to Convert Decimal Degrees to Degrees, Minutes and Seconds

Suppose an angle is given in decimal form, such as:

35.75°

To convert it into degrees, minutes and seconds, follow these steps.

Step 1: Separate the Whole Number

The whole-number part represents the degrees.

For:

35.75°

the degree is:

35°

The decimal part is:

0.75°

Step 2: Convert the Decimal Part into Minutes

Multiply the decimal part by 60:

0.75 × 60 = 45′

Therefore:

35.75° = 35° 45′

Since there is no remaining decimal part, the number of seconds is zero.

So:

35.75° = 35° 45′ 0″

Example 1: Convert 24.5° into Degrees, Minutes and Seconds

Given:

24.5°

Whole-number part:

24°

Decimal part:

0.5°

Convert the decimal part into minutes:

0.5 × 60 = 30′

Therefore:

24.5° = 24° 30′ 0″

Example 2: Convert 18.25° into Degrees, Minutes and Seconds

Given:

18.25°

Whole-number part:

18°

Decimal part:

0.25°

Convert decimal degrees into minutes:

0.25 × 60 = 15′

Therefore:

18.25° = 18° 15′ 0″

Example 3: Convert 42.375° into Degrees, Minutes and Seconds

Given:

42.375°

Whole-number part:

42°

Decimal part:

0.375°

Convert the decimal part into minutes:

0.375 × 60 = 22.5′

So we have:

42° 22.5′

Now separate the whole minutes:

22′

The remaining part is:

0.5′

Convert this remaining part into seconds:

0.5 × 60 = 30″

Therefore:

42.375° = 42° 22′ 30″

General Method for Degree to Minutes and Seconds

To convert a decimal degree into degrees, minutes and seconds:

1. Take the whole-number part as degrees.
2. Multiply the decimal part by 60 to get minutes.
3. Take the whole-number part of the result as minutes.
4. Multiply the remaining decimal part by 60 to get seconds.

Formula

If:

D = x.y°

then:

Degrees = whole-number part of D

Minutes = decimal part of D × 60

If the minutes contain a decimal:

Seconds = decimal part of minutes × 60

Example 4: Convert 73.648° into Degrees, Minutes and Seconds

Given:

73.648°

Degrees:

73°

Decimal part:

0.648

Minutes:

0.648 × 60 = 38.88′

So:

38′

Remaining decimal:

0.88

Seconds:

0.88 × 60 = 52.8″

Therefore:

73.648° = 73° 38′ 52.8″

If the answer is required to the nearest second:

73.648° ≈ 73° 38′ 53″

How to Convert Degrees and Minutes into Decimal Degrees

Sometimes we need to perform the reverse conversion.

Suppose we have:

30° 45′

Since:

1° = 60′

we convert the minutes into degrees by dividing by 60:

45′ = 45/60°

45′ = 0.75°

Therefore:

30° 45′ = 30.75°

Formula

To convert degrees and minutes into decimal degrees:

Decimal degrees = Degrees + Minutes/60

Example 5: Convert 27° 36′ into Decimal Degrees

Given:

27° 36′

Using the formula:

Decimal degrees = 27 + 36/60

= 27 + 0.6

= 27.6°

Therefore:

27° 36′ = 27.6°

How to Convert Degrees, Minutes and Seconds into Decimal Degrees

Suppose the angle is:

40° 30′ 20″

We know:

1° = 60′

and

1° = 3600″

Therefore:

Decimal degrees = Degrees + Minutes/60 + Seconds/3600

For:

40° 30′ 20″

we get:

40 + 30/60 + 20/3600

= 40 + 0.5 + 0.005555…

≈ 40.5056°

Therefore:

40° 30′ 20″ ≈ 40.5056°

How to Convert Minutes and Seconds to Degrees

Minutes and seconds can also be converted directly into degrees.

Minutes to Degrees

Since:

60′ = 1°

we divide minutes by 60:

Degrees = Minutes ÷ 60

For example:

30′ = 30/60° = 0.5°

Seconds to Degrees

Since:

3600″ = 1°

we divide seconds by 3600:

Degrees = Seconds ÷ 3600

For example:

1800″ = 1800/3600° = 0.5°

Important Rules to Remember

Remember these three important relationships:

1° = 60′

1′ = 60″

1° = 3600″

When converting decimal degrees into minutes:

Multiply by 60.

When converting minutes into degrees:

Divide by 60.

When converting minutes into seconds:

Multiply by 60.

When converting seconds into minutes:

Divide by 60.

Common Mistakes Students Should Avoid

Mistake 1: Multiplying by 100

A decimal degree is not converted into minutes by multiplying by 100.

For example:

0.25° ≠ 25′

Instead:

0.25 × 60 = 15′

Therefore:

0.25° = 15′

Mistake 2: Forgetting to Convert the Remaining Decimal

If the result after multiplying by 60 contains a decimal, that decimal represents a fraction of a minute.

For example:

0.375 × 60 = 22.5′

The 22 is minutes, while 0.5 must be converted into seconds:

0.5 × 60 = 30″

Thus:

0.375° = 22′ 30″

Mistake 3: Using 3600 for Minutes

Remember:

1° = 60′

but:

1° = 3600″

So minutes are divided by 60, whereas seconds are divided by 3600 when converting them into degrees.

Short Notes: Degrees, Minutes and Seconds

Basic Relations

1° = 60′

1′ = 60″

1° = 3600″

Decimal Degrees to DMS

1. Whole number = degrees.
2. Decimal part × 60 = minutes.
3. Whole number of minutes = minutes.
4. Remaining decimal × 60 = seconds.

DMS to Decimal Degrees

Decimal degrees = D + M/60 + S/3600

where:

• D = degrees
• M = minutes
• S = seconds

Minutes to Degrees

Degrees = Minutes/60

Seconds to Degrees

Degrees = Seconds/3600

MCQs on Degrees, Minutes and Seconds

1. How many minutes are there in 1 degree?

A. 30
B. 45
C. 60
D. 100

Answer: C. 60

2. How many seconds are there in 1 minute?

A. 30
B. 60
C. 100
D. 360

Answer: B. 60

3. How many seconds are there in 1 degree?

A. 60
B. 600
C. 3600
D. 6000

Answer: C. 3600

4. Convert 0.5° into minutes.

A. 15′
B. 20′
C. 30′
D. 45′

Answer: C. 30′

5. Convert 0.25° into minutes.

A. 10′
B. 15′
C. 20′
D. 25′

Answer: B. 15′

6. Convert 45′ into degrees.

A. 0.25°
B. 0.5°
C. 0.75°
D. 1.25°

Answer: C. 0.75°

7. What is 30° 30′ in decimal degrees?

A. 30.05°
B. 30.25°
C. 30.5°
D. 31°

Answer: C. 30.5°

8. What is 0.5′ in seconds?

A. 15″
B. 20″
C. 30″
D. 60″

Answer: C. 30″

9. Which formula converts DMS into decimal degrees?

A. D + M/60 + S/3600
B. D + M/100 + S/1000
C. D × M × S
D. D + M + S

Answer: A. D + M/60 + S/3600

10. What is 1800″ in degrees?

A. 0.25°
B. 0.5°
C. 1°
D. 2°

Answer: B. 0.5°

Worksheet / Assignment

Part A: Convert Decimal Degrees into Degrees, Minutes and Seconds

Convert the following:

1. 12.5°
2. 25.25°
3. 36.75°
4. 48.125°
5. 62.375°
6. 15.625°
7. 72.45°
8. 84.625°
9. 27.875°
10. 56.325°

Part B: Convert Degrees and Minutes into Decimal Degrees

1. 15° 30′
2. 22° 45′
3. 35° 15′
4. 48° 36′
5. 60° 24′

Part C: Convert Degrees, Minutes and Seconds into Decimal Degrees

1. 10° 20′ 30″
2. 25° 15′ 45″
3. 35° 30′ 20″
4. 42° 12′ 36″
5. 60° 45′ 30″

Part D: Convert Minutes and Seconds into Degrees

1. 30′
2. 45′
3. 90′
4. 1800″
5. 2700″
6. 30″
7. 120″
8. 240″

Challenge Questions

1. Convert 37.5625° into degrees, minutes and seconds.
2. Convert 72° 18′ 45″ into decimal degrees.
3. Convert 5400″ into degrees.
4. Convert 2.75° into degrees, minutes and seconds.
5. Explain why 1 degree contains 3600 seconds.

Conclusion

Converting degree to minutes and seconds is an important mathematical skill. The key relationships are 1° = 60′, 1′ = 60″, and 1° = 3600″. To change decimal degrees into degrees, minutes and seconds, first separate the whole degrees, multiply the decimal part by 60 to obtain minutes, and then convert any remaining decimal part into seconds.

Similarly, degrees, minutes and seconds can be converted back into decimal degrees using:

D + M/60 + S/3600

With regular practice, students can easily perform these conversions and apply them in geometry and trigonometry problems.

For more mathematics notes, recorded lectures, assessments, lesson plans and other educational material, visit Nisar Math Academy.

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