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Radians and Conversion
Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn has arc 2 pi r, so 2 pi rad = 360 deg. Half of that is pi rad = 180 deg. To convert, rad to deg multiply by 180/pi, and deg to rad multiply by pi/180. Then 45 deg = pi/4 and 150 deg = 5pi/6.
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Radians and Conversion
What is it?
Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn has arc 2 pi r, so 2 pi rad = 360 deg. Half of that is pi rad = 180 deg. To convert, rad to deg multiply by 180/pi, and deg to rad multiply by pi/180. Then 45 deg = pi/4 and 150 deg = 5pi/6.
In simple words
- Circular measure uses radians.
- On a circle, theta = arc/radius.
- Rad to deg: multiply by 180/pi.
- Because pi rad = 180 deg, a degree is pi/180 of a radian.
Important terms
theta = arc/radius: The radian measure of a central angle is theta = arc/radius.
1 radian: You get 1 radian when the arc length equals the radius.
2 pi rad = 360 deg: A full turn has arc 2 pi r, so 2 pi rad = 360 deg.
pi rad = 180 deg: Half of that is pi rad = 180 deg.
What you should learn
- Explain radians and conversion in clear mathematical language
- Apply the ideas in what a radian is
- Solve problems involving converting
Examples
- arc = radius gives 1 radian
- full turn 2 pi rad = 360 deg
- straight angle pi rad = 180 deg
WHAT A RADIAN IS
On a circle, theta = arc/radius. If the arc is as long as the radius, theta is 1 radian. The radius unit cancels, so a radian is a pure number, but we still write rad to show the system. A full turn wraps an arc of 2 pi r, hence 2 pi rad = 360 deg. Cutting the turn in half gives pi rad = 180 deg, the conversion anchor you should memorise.
CONVERTING
Rad to deg: multiply by 180/pi. Deg to rad: multiply by pi/180. Worked: 45 deg = 45 pi/180 = pi/4, so 45 deg = pi/4. Worked: 150 deg = 150 pi/180 = 5pi/6, so 150 deg = 5pi/6. Reverse: pi/6 rad = 30 deg because (pi/6)*(180/pi)=30. Cancel pi when it appears on top and bottom. Keep the factor 180 next to pi so you do not invert the rule.
Recommended for Class 9
Full school-level explanation
Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn has arc 2 pi r, so 2 pi rad = 360 deg. Half of that is pi rad = 180 deg. To convert, rad to deg multiply by 180/pi, and deg to rad multiply by pi/180. Then 45 deg = pi/4 and 150 deg = 5pi/6.
One full turn, two unit systems:
What it means
Circular measure uses radians.
Main ideas
1. WHAT A RADIAN IS: On a circle, theta = arc/radius. 2. CONVERTING: Rad to deg: multiply by 180/pi.
Why it matters
Because pi rad = 180 deg, a degree is pi/180 of a radian. Do not treat 1 rad as 1 deg; 1 rad is about 57 deg. Leave exact answers in terms of pi unless the question asks for a decimal.
How to answer in an exam
The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
WHAT A RADIAN IS
On a circle, theta = arc/radius. If the arc is as long as the radius, theta is 1 radian. The radius unit cancels, so a radian is a pure number, but we still write rad to show the system. A full turn wraps an arc of 2 pi r, hence 2 pi rad = 360 deg. Cutting the turn in half gives pi rad = 180 deg, the conversion anchor you should memorise.
Remember
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
- pi rad = 180 deg
Examples
- arc = radius gives 1 radian
- full turn 2 pi rad = 360 deg
- straight angle pi rad = 180 deg
CONVERTING
Rad to deg: multiply by 180/pi. Deg to rad: multiply by pi/180. Worked: 45 deg = 45 pi/180 = pi/4, so 45 deg = pi/4. Worked: 150 deg = 150 pi/180 = 5pi/6, so 150 deg = 5pi/6. Reverse: pi/6 rad = 30 deg because (pi/6)*(180/pi)=30. Cancel pi when it appears on top and bottom. Keep the factor 180 next to pi so you do not invert the rule.
Remember
- 180/pi
- pi/180
- 45 deg = pi/4
- 150 deg = 5pi/6
Examples
- 45 deg = pi/4
- 150 deg = 5pi/6
- pi/3 rad = 60 deg
Remember
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
- pi rad = 180 deg
- 180/pi
- pi/180
- 45 deg = pi/4
- 150 deg = 5pi/6
How the ideas connect
Because pi rad = 180 deg, a degree is pi/180 of a radian. Do not treat 1 rad as 1 deg; 1 rad is about 57 deg. Leave exact answers in terms of pi unless the question asks for a decimal.
Deeper understanding
Go beyond memorising Radians and Conversion. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Because pi rad = 180 deg, a degree is pi/180 of a radian. Do not treat 1 rad as 1 deg; 1 rad is about 57 deg. Leave exact answers in terms of pi unless the question asks for a decimal.
Common mistakes
Exam focus
Core Idea
Key Terms
Main Pattern
Worked Reminder
Exam focus
Quick revision
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
- pi rad = 180 deg
- theta = arc/radius: The radian measure of a central angle is theta = arc/radius.
- 1 radian: You get 1 radian when the arc length equals the radius.
The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
For long answers on Radians and Conversion: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
- pi rad = 180 deg
Important formulas
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
Common exam traps
- The easy memory is pi rad = 180 deg.
- From that, rad to deg 180/pi and deg to rad pi/180.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The easy memory is pi rad = 180 deg.
- Mistake
From that, rad to deg 180/pi and deg to rad pi/180.
- Mistake
Do not mix up what a radian is with converting; use the condition given in the question.
- Mistake
The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
theta = arc/radius
1 radian
2 pi rad = 360 deg
pi rad = 180 deg
Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn
theta = arc/radius
- Definition
- Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn
- Formula
- theta = arc/radius
- Example
- arc = radius gives 1 radian
- Critical warning
- The easy memory is pi rad = 180 deg.
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Exam Practice
Define
Q1. Define Radians and Conversion.
Explain
Q2. Explain: WHAT A RADIAN IS.
Example
Q3. Give one clear example of Radians and Conversion and say why it fits.
Mistake
Q4. State one common mistake students make about Radians and Conversion.
Exam note
Q5. Write a short exam note on Radians and Conversion: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.