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Degrees and radians

Trigonometry · 23 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Write the rule
  2. Multiply
  3. Make it simpler
  4. Check

The steps for Convert 30° to radians. Every solution names its steps, so you always know what comes next.

Worked examples

Convert 30° to radians Easy

The question

Question: change 30° to radians.

1 Write the rule

Radians are another way to measure an angle.

A half turn is 180°.

In radians, a half turn is π.

So 180° = π.

180° = π

We divide both sides by 180.

One degree is π/180.

1° = π180

Careful!

To get radians, π must end up in the answer.

So π goes on top.

Careful: to radians: × π180

2 Multiply

30° is 30 times one degree.

So we multiply 30 by π/180.

30° = 30 × π180

Top times top: 30π.

The bottom stays 180.

30° = 30π180

3 Make it simpler

30 and 180 can both be divided by 30.

It is the biggest number that goes into both.

30 ÷ 30 is 1.

180 ÷ 30 is 6.

30 ÷ 30 = 1
180 ÷ 30 = 6

So 30° = π/6 radians.

30° = π6

4 Check

Let's check: go back to degrees.

Times 180° over π: the πs cancel.

π6 × 180°π = 180°6 = 30°

We get 30° again: the angle we started with.

So it is right!

Let's look back.

First: write the rule.

Then: multiply.

Then: make it simpler.

Last: check.

Answer
30° = π6

▶ Watch it solved

Convert 120° to radians Medium

The question

Question: change 120° to radians.

1 Write the rule

Radians are another way to measure an angle.

A half turn is 180°.

In radians, a half turn is π.

So 180° = π.

180° = π

We divide both sides by 180.

One degree is π/180.

1° = π180

Careful!

To get radians, π must end up in the answer.

So π goes on top.

Careful: to radians: × π180

2 Multiply

120° is 120 times one degree.

So we multiply 120 by π/180.

120° = 120 × π180

Top times top: 120π.

The bottom stays 180.

120° = 120π180

3 Make it simpler

120 and 180 can both be divided by 60.

It is the biggest number that goes into both.

120 ÷ 60 is 2.

180 ÷ 60 is 3.

120 ÷ 60 = 2
180 ÷ 60 = 3

So 120° = 2π/3 radians.

120° = 2π3

4 Check

Let's check: go back to degrees.

Times 180° over π: the πs cancel.

2π3 × 180°π = 2 × 180°3 = 120°

We get 120° again: the angle we started with.

So it is right!

Let's look back.

First: write the rule.

Then: multiply.

Then: make it simpler.

Last: check.

Answer
120° = 2π3

▶ Watch it solved

Convert -45° to radians Harder

The question

Question: change −45° to radians.

1 Write the rule

Radians are another way to measure an angle.

A half turn is 180°.

In radians, a half turn is π.

So 180° = π.

180° = π

We divide both sides by 180.

One degree is π/180.

1° = π180

Careful!

To get radians, π must end up in the answer.

So π goes on top.

Careful: to radians: × π180

2 Multiply

−45° is −45 times one degree.

So we multiply −45 by π/180.

−45° = −45 × π180

Top times top: 45π.

The bottom stays 180.

−45° = −45π180

3 Make it simpler

45 and 180 can both be divided by 45.

It is the biggest number that goes into both.

45 ÷ 45 is 1.

180 ÷ 45 is 4.

45 ÷ 45 = 1
180 ÷ 45 = 4

So −45° = −π/4 radians.

−45° = −π4

4 Check

Let's check: go back to degrees.

Times 180° over π: the πs cancel.

−π4 × 180°π = −180°4 = −45°

We get −45° again: the angle we started with.

So it is right!

Let's look back.

First: write the rule.

Then: multiply.

Then: make it simpler.

Last: check.

Answer
−45° = −π4

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. 30° to radians
    Answer
    30° = π6
  2. 45° to radians
    Answer
    45° = π4
  3. 60° to radians
    Answer
    60° = π3
  4. 90° to radians
    Answer
    90° = π2
  5. 360° to radians
    Answer
    360° = 2π
  6. π/2 to degrees
    Answer
    π2 = 90°
  7. π/3 to degrees
    Answer
    π3 = 60°
  8. π/6 to degrees
    Answer
    π6 = 30°

Medium

  1. 120° to radians
    Answer
    120° = 2π3
  2. 135° to radians
    Answer
    135° = 3π4
  3. 210° to radians
    Answer
    210° = 7π6
  4. 300° to radians
    Answer
    300° = 5π3
  5. 240° to radians
    Answer
    240° = 4π3
  6. 3π/4 to degrees
    Answer
    3π4 = 135°
  7. 7π/6 to degrees
    Answer
    7π6 = 210°
  8. 5π/3 to degrees
    Answer
    5π3 = 300°
  9. 11π/6 to degrees
    Answer
    11π6 = 330°

Harder

  1. -45° to radians
    Answer
    −45° = −π4
  2. -120° to radians
    Answer
    −120° = −2π3
  3. 450° to radians
    Answer
    450° = 5π2
  4. 5π/12 to degrees
    Answer
    5π12 = 75°
  5. 2 radians to degrees
    Answer
    2 radians ≈ 114.6°
  6. 3 radians to degrees
    Answer
    3 radians ≈ 171.9°