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Amy Ferguson Moncure

IB AA · Unit 2 · 18A Radian Measure

Convert Degrees to Radians

Write a fraction. Put π radians over 180° so the degrees cancel. Multiply straight across, then reduce.

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Key Ideas Easy Medium Hard

π radians180°

Then reduce.

What to Remember

The conversion

1 full turn is 360° and also 2π radians. That gives the factor

π radians180°

Always put degrees on the bottom so they cancel. Multiply straight across, then reduce.

Three significant figures

When the angle is not a friendly fraction of 180°, reduce the fraction of π, then use a calculator.

Round the decimal to 3 significant figures.

Easy

In terms of π

Friendly angles. Reduce the fraction.

Example
Convert 240° to radians. Give the answer in terms of π.

240°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

240π180 = 4π3

Answer: 4π3

1
Convert 30° to radians. Give the answer in terms of π.
Show Answer

30°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

30π180 = π6

Answer: π6

2
Convert 135° to radians. Give the answer in terms of π.
Show Answer

135°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

135π180 = 3π4

Answer: 3π4

3
Convert 180° to radians. Give the answer in terms of π.
Show Answer

180°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

180π180 = π

Answer: π

4
Convert 225° to radians. Give the answer in terms of π.
Show Answer

225°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

225π180 = 5π4

Answer: 5π4

5
Convert 315° to radians. Give the answer in terms of π.
Show Answer

315°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

315π180 = 7π4

Answer: 7π4

Medium

Still exact

These reduce, but not all the way to a single π.

Example
Convert 330° to radians. Give the answer in terms of π.

330°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

330π180 = 11π6

Answer: 11π6

1
Convert 20° to radians. Give the answer in terms of π.
Show Answer

20°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

20π180 = π9

Answer: π9

2
Convert 75° to radians. Give the answer in terms of π.
Show Answer

75°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

75π180 = 5π12

Answer: 5π12

3
Convert 80° to radians. Give the answer in terms of π.
Show Answer

80°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

80π180 = 4π9

Answer: 4π9

4
Convert 105° to radians. Give the answer in terms of π.
Show Answer

105°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

105π180 = 7π12

Answer: 7π12

5
Convert 200° to radians. Give the answer in terms of π.
Show Answer

200°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

200π180 = 10π9

Answer: 10π9

Hard

Three significant figures

Reduce first, then use a calculator.

Example
Convert 22° to radians. Round to 3 significant figures.

22°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

22π180 = 11π90

Use a calculator. Round to 3 significant figures.

11π90 ≈ 0.384 radians

Answer: 0.384 radians

1
Convert 17° to radians. Round to 3 significant figures.
Show Answer

17°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across. Nothing cancels.

The answer is 17π180.

Use a calculator. Round to 3 significant figures.

17π180 ≈ 0.297 radians

Answer: 0.297 radians

2
Convert 41° to radians. Round to 3 significant figures.
Show Answer

41°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across. Nothing cancels.

The answer is 41π180.

Use a calculator. Round to 3 significant figures.

41π180 ≈ 0.716 radians

Answer: 0.716 radians

3
Convert 53° to radians. Round to 3 significant figures.
Show Answer

53°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across. Nothing cancels.

The answer is 53π180.

Use a calculator. Round to 3 significant figures.

53π180 ≈ 0.925 radians

Answer: 0.925 radians

4
Convert 128° to radians. Round to 3 significant figures.
Show Answer

128°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

128π180 = 32π45

Use a calculator. Round to 3 significant figures.

32π45 ≈ 2.23 radians

Answer: 2.23 radians

5
Convert 214° to radians. Round to 3 significant figures.
Show Answer

214°1×π rad180°

Degrees go on the bottom of the conversion so the degrees cancel.

Multiply straight across, then reduce.

214π180 = 107π90

Use a calculator. Round to 3 significant figures.

107π90 ≈ 3.74 radians

Answer: 3.74 radians

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