MathematicsChapter 6: Trigonometry and Bearing
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 (1/2) r^2 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, convert first: thetarad = thetadeg x pi / 180.
Mathematical form
thetarad = thetadeg x pi / 180. Worked
r = 10 cm
angle = 60 deg. theta
Arc = 10 pi/3 cm. Sector
Relationship
- arc = r theta
- sector = (12) r2 theta
Overview
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.
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.
Examples r = 6, theta = pi/3 -> area = 6 pi · Pizza slice is a sector · Do not use degrees inside r theta unless converted
Worked example
- 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.
In simple words
The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
Key points
- arc = r theta
- sector = (12) r2 theta
- radians
- fraction of the circle
- same units
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Arc Length and Sector Area
Sticky Notes · Quick Revision
Core Idea
- 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) r2 theta.
Key Points
- arc = r theta
- sector = (12) r2 theta
- radians
Simple Check
- Explain arc length and sector area in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- The formula arc = r theta is illegal if theta is still in degrees. Convert, or use (theta/360) x 2 pi r.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- arc = r theta
- sector = (12) r2 theta
- radians
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