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Angle worksheet

Grades 10–11 · CCSS HSG.C.B.5, HSF.TF.A.1

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Conversion: DMS to decimal degrees
- Degrees: 40 °
- Minutes: 28 ′
- Seconds: 54 ″
- Negative angle (south latitude or west longitude): No
Find: Decimal degrees
Answer
2.
Problem 2.
- Conversion: DMS to decimal degrees
- Degrees: 32 °
- Minutes: 26 ′
- Seconds: 40 ″
- Negative angle (south latitude or west longitude): No
Find: Decimal degrees
Answer
3.
Problem 3.
- Conversion: Between angle units
- Angle: 180
- From: Radians (rad)
- To: Turns (turn)
Find: In turns
Answer
4.
Problem 4.
- Conversion: DMS to decimal degrees
- Degrees: 50 °
- Minutes: 21 ′
- Seconds: 46 ″
- Negative angle (south latitude or west longitude): No
Find: Decimal degrees
Answer
5.
Problem 5.
- Conversion: Between angle units
- Angle: 225
- From: Degrees (°)
- To: Turns (turn)
Find: In turns
Answer
6.
Problem 6.
- Conversion: DMS to decimal degrees
- Degrees: 45 °
- Minutes: 36 ′
- Seconds: 38 ″
- Negative angle (south latitude or west longitude): No
Find: Decimal degrees
Answer
7.
Problem 7.
- Conversion: Decimal degrees to DMS
- Decimal degrees: -92.5 °
Find: Degrees, minutes, seconds
Answer
8.
Problem 8.
- Conversion: Decimal degrees to DMS
- Decimal degrees: -90 °
Find: Degrees, minutes, seconds
Answer
9.
Problem 9.
- Conversion: DMS to decimal degrees
- Degrees: 32 °
- Minutes: 22 ′
- Seconds: 42.5 ″
- Negative angle (south latitude or west longitude): No
Find: Decimal degrees
Answer
10.
Problem 10.
- Conversion: Decimal degrees to DMS
- Decimal degrees: -65 °
Find: Degrees, minutes, seconds
Answer

Answer key: Angle worksheet

1. Decimal degrees: 40.48 °
1. Convert minutes and seconds to fractions of a degree: a degree has 60 minutes and 3,600 seconds
  $40 + \frac{28}{60} + \frac{54}{3,600}$ $= 40 + 0.4666666667 + 0.015$
2. Add the degrees and those fractions to get decimal degrees
  $40 + 0.4666666667 + 0.015$ $= 40.4816666667 °$
2. Decimal degrees: 32.44 °
1. Convert minutes and seconds to fractions of a degree: a degree has 60 minutes and 3,600 seconds
  $32 + \frac{26}{60} + \frac{40}{3,600}$ $= 32 + 0.4333333333 + 0.0111111111$
2. Add the degrees and those fractions to get decimal degrees
  $32 + 0.4333333333 + 0.0111111111$ $= 32.4444444444 °$
3. In turns: 28.65 turn
1. Convert to degrees first, since every unit has a fixed size in degrees: 1 rad = 180/π ≈ 57.2957795131°, so multiply
  $180 rad \times 57.2957795131$ $= 10,313.2403123548 °$
2. Convert to turns: 1 turn = 360°, so divide
  $\frac{10,313.2403123548 °}{360}$ $= 28.6478897565 turn$
4. Decimal degrees: 50.36 °
1. Convert minutes and seconds to fractions of a degree: a degree has 60 minutes and 3,600 seconds
  $50 + \frac{21}{60} + \frac{46}{3,600}$ $= 50 + 0.35 + 0.0127777778$
2. Add the degrees and those fractions to get decimal degrees
  $50 + 0.35 + 0.0127777778$ $= 50.3627777778 °$
5. In turns: 0.63 turn
1. Convert to turns: 1 turn = 360°, so divide
  $\frac{225 °}{360}$ $= 0.625 turn$
6. Decimal degrees: 45.61 °
1. Convert minutes and seconds to fractions of a degree: a degree has 60 minutes and 3,600 seconds
  $45 + \frac{36}{60} + \frac{38}{3,600}$ $= 45 + 0.6 + 0.0105555556$
2. Add the degrees and those fractions to get decimal degrees
  $45 + 0.6 + 0.0105555556$ $= 45.6105555556 °$
7. Degrees, minutes, seconds: −92° 30′ 0″
1. The whole number is the degrees. Keep the leftover fraction for the next step
  $92.5 °$ $= 92 ° + 0.5 °$
2. Multiply the leftover fraction by 60, because a degree has 60 minutes. The whole number is the minutes
  $0.5 \times 60$ $= 30 ′$
3. Multiply the leftover fraction of a minute by 60 to get the seconds, rounded to 4 decimals
  $0 \times 60$ $= 0 ″$
4. Put the pieces together and keep the negative sign
  $−92° 30′ 0″$
8. Degrees, minutes, seconds: −90° 0′ 0″
1. The whole number is the degrees. Keep the leftover fraction for the next step
  $90 °$ $= 90 ° + 0 °$
2. Multiply the leftover fraction by 60, because a degree has 60 minutes. The whole number is the minutes
  $0 \times 60$ $= 0 ′$
3. Multiply the leftover fraction of a minute by 60 to get the seconds, rounded to 4 decimals
  $0 \times 60$ $= 0 ″$
4. Put the pieces together and keep the negative sign
  $−90° 0′ 0″$
9. Decimal degrees: 32.38 °
1. Convert minutes and seconds to fractions of a degree: a degree has 60 minutes and 3,600 seconds
  $32 + \frac{22}{60} + \frac{42.5}{3,600}$ $= 32 + 0.3666666667 + 0.0118055556$
2. Add the degrees and those fractions to get decimal degrees
  $32 + 0.3666666667 + 0.0118055556$ $= 32.3784722222 °$
10. Degrees, minutes, seconds: −65° 0′ 0″
1. The whole number is the degrees. Keep the leftover fraction for the next step
  $65 °$ $= 65 ° + 0 °$
2. Multiply the leftover fraction by 60, because a degree has 60 minutes. The whole number is the minutes
  $0 \times 60$ $= 0 ′$
3. Multiply the leftover fraction of a minute by 60 to get the seconds, rounded to 4 decimals
  $0 \times 60$ $= 0 ″$
4. Put the pieces together and keep the negative sign
  $−65° 0′ 0″$