Secondary Mathematics · Worked Repair Guide 37
Radian measure becomes much easier when it is understood as a ratio rather than a mysterious alternative to degrees. One radian is the central angle that subtends an arc equal in length to the radius. From that definition, the formulas for arc length and sector area become direct relationships instead of extra rules to memorise.
This guide develops one central habit: identify the angle unit before using a circular-measure formula. The simple forms s=rθ and A=1/2r²θ require θ in radians. Degree-based fraction-of-a-circle formulas remain valid in degrees. Mixing the systems produces wrong answers that can look numerically plausible.
All diagrams and values below are original teaching constructions. This Secondary worked-repair page complements the existing circle-geometry guide and remains distinct from the deeper Additional Mathematics radian/trigonometric synthesis route.
1. One full turn is 2π radians
A full circumference has length 2πr. Since one radian corresponds to arc length r, the number of radians in one full turn is 2πr/r=2π.
Therefore 360°=2π radians and 180°=π radians.
Entry check: 90°=π/2 radians and 45°=π/4 radians.
These anchor values make most degree-radian conversions a proportion rather than a fresh calculation.
2. Convert degrees to radians by multiplying by π/180
For an angle d°:
θ radians=d×π/180.
Example: 150°=150π/180=5π/6 radians.
Example: 72°=72π/180=2π/5 radians.
Keep exact multiples of π unless a decimal radian value is specifically requested.
3. Convert radians to degrees by multiplying by 180/π
For θ radians:
d°=θ×180/π.
Example: 7π/12 radians=7π/12×180/π=105°.
Example: 2.4 radians≈137.5°.
When π cancels exactly, show the cancellation rather than converting π prematurely to 3.142.
4. Arc length in radians is s=rθ
The definition of radian gives θ=s/r, so s=rθ.
If r=6 cm and θ=2 radians, s=12 cm.
If r=9 cm and θ=π/3, s=3π cm.
The formula is linear in both radius and angle. Doubling r with θ fixed doubles the arc; doubling θ with r fixed also doubles the arc.
5. The degree formula and radian formula describe the same arc
In degrees, arc length=(d/360)×2πr.
Since d=180θ/π, substitute:
(180θ/π)/360×2πr=rθ.
The radian formula is not a different geometry. It is the degree formula simplified by using an angle unit naturally tied to arc length.
6. Sector area in radians is A=1/2r²θ
A full circle has area πr² and angle 2π radians. A sector of angle θ therefore occupies fraction θ/(2π) of the full circle.
Area=θ/(2π)×πr²=1/2r²θ.
If r=8 cm and θ=π/4, area=1/2×64×π/4=8π cm².
If θ is in degrees, do not use this form without conversion.
7. Sector perimeter includes the arc and two radii
For radius r and radian angle θ, sector perimeter P=rθ+2r.
Example: r=5, θ=1.2 gives arc length6 and perimeter16.
Do not report arc length alone when the question asks for the perimeter of the sector.
Trace the boundary: radius, arc, radius.
8. Reverse arc-length problems solve for radius or angle
If s=15 cm and θ=2.5 radians, r=s/θ=6 cm.
If s=14 cm and r=8 cm, θ=s/r=1.75 radians.
Before solving, confirm that the angle variable is actually in radians.
Units also provide a check: arc length divided by radius is dimensionless, as an angle in radians should be.
9. Reverse sector-area problems also require radians
If sector area=40 cm², r=5 cm and θ is unknown:
40=1/2×25×θ, so θ=80/25=3.2 radians.
If area=18π cm² and θ=π/2, then18π=1/2r²(π/2)=πr²/4, so r²=72 and r=6√2 cm.
Lengths use the positive root.
10. A circular segment is sector minus triangle
A minor segment is the region between a chord and its corresponding minor arc.
For central angle θ, sector area=1/2r²θ.
The isosceles triangle formed by the two radii has area=1/2r² sinθ.
Therefore minor segment area=1/2r²(θ−sinθ), with θ in radians.
11. Work a segment-area example
Let r=10 cm and θ=π/3.
Sector area=1/2×100×π/3=50π/3.
Triangle area=1/2×100×sin60°=50×√3/2=25√3.
Minor segment area=50π/3−25√3≈9.06 cm².
The segment is smaller than the sector, as expected.
12. The major segment is the rest of the circle
If the minor segment area is S and full circle area is πr², major segment area=πr²−S.
For the previous example, major segment area=100π−(50π/3−25√3)=250π/3+25√3.
Do not use the minor-segment subtraction picture for a major segment without accounting for the rest of the circle.
Sketch which region is being requested before calculating.
13. Chord length can be connected to the central angle
For a circle radius r and minor central angle θ, splitting the isosceles triangle in half gives chord length c=2r sin(θ/2).
If r=6 and θ=π/3, c=12 sin(π/6)=6.
This is a trigonometric extension that is useful when segment perimeter or chord constraints appear.
Match this extension to the learner’s actual syllabus depth.
14. Segment perimeter combines a chord and an arc
The perimeter of a minor segment is chord length + minor arc length.
For r=6 and θ=π/3, chord=6 and arc=6×π/3=2π.
Segment perimeter=6+2π.
This differs from sector perimeter, which would include two radii instead of the chord.
15. Calculator angle mode is part of the problem state
If θ is measured in radians and sinθ is required, the calculator must interpret the input in radians.
For example, sin(π/6)=0.5 in radian mode.
Typing the decimal0.524 while the calculator is in degree mode evaluates sin(0.524°), not sin(0.524 radians).
Always check the angle unit before trigonometric work inside circular-measure questions.
16. Small radian angles reveal useful approximations
For small positive θ measured in radians, sinθ is close to θ.
Therefore minor segment area 1/2r²(θ−sinθ) becomes very small compared with the sector.
This is a qualitative check: a very narrow sector should have its chord close to its arc and its segment area small.
No calculus is required to use this as a plausibility observation.
17. Composite circular regions require boundary discipline
Suppose a shape is formed by two sectors sharing a radius line. Area calculations add or subtract the relevant regions; perimeter calculations include only the exposed outer boundary.
An internal radius used to split the figure is not part of the external perimeter.
This is the circular version of the composite-solid principle from mensuration.
Trace the requested boundary once before writing a perimeter expression.
18. Capstone: sector, triangle, segment and perimeter
A circle has radius12 cm and central angle θ=2π/3.
Arc length=12×2π/3=8π cm.
Sector area=1/2×144×2π/3=48π cm².
Triangle area=1/2×144×sin120°=72×√3/2=36√3 cm².
Minor segment area=48π−36√3 cm².
Chord length=24 sin60°=12√3 cm, so segment perimeter=8π+12√3 cm.
19. Independent practice
- Convert60° to radians.
- Convert225° to radians.
- Convert5π/6 radians to degrees.
- Convert2.2 radians to degrees, approximately.
- Find arc length for r=7, θ=1.4.
- Find arc length for r=9, θ=π/3.
- Find sector area for r=8, θ=π/4.
- Find sector perimeter for r=5, θ=1.2.
- Arc length15 and angle2.5 radians are given. Find r.
- Arc length14 and radius8 are given. Find θ.
- Sector area40 and radius5 are given. Find θ.
- Sector area18π and θ=π/2. Find r.
- For r=10, θ=π/3, find the minor segment area in exact form.
- Find the chord for r=6, θ=π/3.
- Find the minor segment perimeter for Question14.
- Explain why s=rθ requires θ in radians.
- Explain why sector perimeter is not just the arc length.
- State the full-circle angle in radians.
- For r=12, θ=2π/3, find sector area.
- For the same data, find the exact minor segment area.
20. Worked answers
1. π/3.
2. 5π/4.
3. 150°.
4. Approximately126.1°.
5. 9.8.
6. 3π.
7. 8π.
8. 16. Arc6 plus two radii10.
9. 6.
10. 1.75 radians.
11. 3.2 radians.
12. 6√2.
13. 50π/3−25√3.
14. 6.
15. 6+2π.
16. Radian measure is defined by θ=s/r; the simplified formula comes directly from that definition.
17. The sector boundary contains the arc and two radii.
18. 2π.
19. 48π.
20. 48π−36√3.
21. Diagnose circular-measure errors by unit first
Common failures include using s=rθ with degrees, forgetting the two radii in sector perimeter, subtracting the wrong triangle when finding a segment, confusing chord and arc, or leaving the calculator in degree mode while evaluating a radian sine.
A useful repair note says “convert angle first”, “radian means arc/radius”, “segment=sector−triangle”, or “trace the requested boundary”.
Then give the same geometry once in degrees and once in radians. A secure learner should produce the same physical arc or sector after correct conversion.
22. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Circle Geometry — Arcs, Sectors, Chords and Tangents for theorem and degree-based foundations and Trigonometry, Pythagoras, Bearings and Elevation for sine/cosine foundations.
Within Batch10, continue to Graphical Solutions: Equations, Intersections and Roots, Three-Dimensional Trigonometry and Spatial Diagrams, or Plane Mensuration: Parallelograms, Trapezia and Composite Figures.
23. Sources and scope
The circular figures, calculations and practice questions are original teaching material. Radian definitions and circular-measure formulas are used in their standard Euclidean sense.
For current Singapore Secondary curriculum and assessment scope, consult the relevant MOE/SEAB syllabus for the learner’s subject level and year. Match chord and segment extensions to the actual course being studied.
Complete Secondary Mathematics repair route: browse all 48 worked repair guides or return to the Secondary Mathematics Learning Hub.