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Arc Length and Sector Area
On a circle of radius r, a central angle theta in radians cuts an arc of length r theta and a sector of area (12) r^2 theta. If the angle is in degrees, convert first: thetarad = thetadeg x pi / 180.
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Arc Length and Sector Area
What is it?
On a circle of radius r, a central angle theta in radians cuts an arc of length r theta and a sector of area (12) r^2 theta. If the angle is in degrees, convert first: thetarad = thetadeg x pi / 180.
In simple words
Arc Length and Sector Area is a core idea in this chapter. Learn the meaning first, then the main parts, then one clear example.
Examples
- r = 7, theta = pi/2 -> arc = 7pi/2
- 90 deg = pi/2 rad
- theta = 1 rad, r = 10 -> arc = 10
ARC LENGTH
Think of the full circumference 2 pi r as the arc for angle 2 pi. A fraction theta / (2 pi) of the circle gives arc = (theta / 2 pi) x 2 pi r = r theta. Units: r and arc in the same length unit.
SECTOR AREA
Full disk area is pi r^2. The same fraction theta / (2 pi) gives area = (12) r^2 theta. A segment is the bit between a chord and its arc; that is extra, not the sector.
Recommended for Class 9
Full school-level explanation
On a circle of radius r, a central angle theta in radians cuts an arc of length r theta and a sector of area (12) r^2 theta. If the angle is in degrees, convert first: thetarad = thetadeg x pi / 180.
Two conversions you must not skip:
ARC LENGTH
Think of the full circumference 2 pi r as the arc for angle 2 pi. A fraction theta / (2 pi) of the circle gives arc = (theta / 2 pi) x 2 pi r = r theta. Units: r and arc in the same length unit.
Remember
- fraction of the circle
- same units
Examples
- r = 7, theta = pi/2 -> arc = 7pi/2
- 90 deg = pi/2 rad
- theta = 1 rad, r = 10 -> arc = 10
SECTOR AREA
Full disk area is pi r^2. The same fraction theta / (2 pi) gives area = (12) r^2 theta. A segment is the bit between a chord and its arc; that is extra, not the sector.
Remember
- fraction of the disk
- sector not segment
Examples
- r = 6, theta = pi/3 -> area = 6 pi
- Pizza slice is a sector
- Do not use degrees inside r theta unless converted
Remember
- arc = r theta
- sector = (12) r2 theta
- radians
- convert degrees first
- radians or the 360 formula
- fraction of the circle
- same units
- fraction of the disk
How the ideas connect
Worked: r = 10 cm, angle = 60 deg. theta = 60 x pi/180 = pi/3. Arc = 10 pi/3 cm. Sector = (12)(100)(pi/3) = 50 pi/3 cm2.
Deeper understanding
Go beyond memorising Arc Length and Sector Area. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: r = 10 cm, angle = 60 deg. theta = 60 x pi/180 = pi/3. Arc = 10 pi/3 cm. Sector = (12)(100)(pi/3) = 50 pi/3 cm2.
Exam focus
Quick revision
- arc = r theta
- sector = (12) r2 theta
- radians
- convert degrees first
- ARC LENGTH: Think of the full circumference 2 pi r as the arc for angle 2 pi. A fraction theta / (2 pi) of the c
- SECTOR AREA: Full disk area is pi r^2. The same fraction theta / (2 pi) gives area = (12) r^2 theta. A segment i
The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
For long answers on Arc Length and Sector Area: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- arc = r theta
- sector = (12) r2 theta
- radians
- convert degrees first
Important formulas
- arc = r theta
- sector = (12) r2 theta
- radians
Common exam traps
- The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
- The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
- Mistake
The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
- Mistake
The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
arc = r theta
sector = (12) r2 theta
radians
convert degrees first
On a circle of radius r, a central angle theta in radians cuts an arc of length r theta and a sector of area (12) r^2 theta. If the angle is in degrees, conver
arc = r theta
- Definition
- On a circle of radius r, a central angle theta in radians cuts an arc of length r theta and a sector of area (12) r^2 theta. If the angle is in degrees, conver
- Formula
- arc = r theta
- Example
- r = 7, theta = pi/2 -> arc = 7pi/2
- Critical warning
- The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
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Exam Practice
Define
Q1. Define Arc Length and Sector Area.
Explain
Q2. Explain: ARC LENGTH.
Example
Q3. Give one clear example of Arc Length and Sector Area and say why it fits.
Mistake
Q4. State one common mistake students make about Arc Length and Sector Area.
Exam note
Q5. Write a short exam note on Arc Length and Sector Area: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.