Calculate curved lengths and areas. Work backwards to find a sector.
HL: measure angles in radians.
Open your notebook. Have your calculator ready.
Use ← / → to move between slides. Answers stay hidden until you open a check.
On your own · 5 minutes
Start with what you know.
Circles
Write expressions for the circumference and area of a circle with radius r.
What fraction of a full turn is 120°?
Triangles and algebra
Two sides of a triangle are 8 cm and 10 cm. The included angle is 30°. Find its area.
Solve x² − 9x + 20 = 0.
Think first · Then compare with a partner
Two sector-shaped gardens.
Garden A
Garden B
Which has the longer curved boundary? Which has the greater area?
Write your predictions and a reason. Do not calculate yet.
Work with a partner · Show your calculations
Use fractions of whole circles.
Garden A: r = 6 m, θ = 120°. Garden B: r = 8 m, θ = 90°.
Garden
Fraction of a circle
Curved boundary / m
Area / m²
A
…
…
…
B
…
…
…
Keep exact answers in terms of π. Compare your results with your predictions.
Write and explain
What fraction of the circle?
A sector has radius r and central angle θ°.
Fraction of circle = θ360
Use this fraction to write a formula for: 1. arc length s; 2. sector area A.
Explain why the same fraction appears in both formulas.
Sketch and label
Arc length is not the perimeter.
Trace the entire boundary of a sector.
P = r + s + r = 2r + s
A minor sector has an angle below 180°. A major sector has an angle above 180° and below 360°.
Do both sectors have two straight edges? Explain what must be added to the arc.
Minor sectorMajor sector
Here, each sector has an angle below one full turn. A complete disk has perimeter 2πr.
On your own · 2 minutes
Find the missing angle.
A sector has radius 6 cm and arc length 3π cm.
Find its total perimeter.
Find its central angle in degrees.
s = θ360 × 2πr
Use the measurements, not the appearance of the sketch.
Work with a partner · 8 minutes
Can you find the area without the angle?
You know the radius r and arc length s.
s = θ360 × 2πr
A = θ360 × πr²
Find a formula for A using only r and s. Write a derivation that another pair can follow.
Predict · Then test
Same arc. Different radius.
A = ½rs
Keep s = 4π m, as in both gardens.
A third garden has radius 12 m. Predict its area without finding the angle.
Then move the slider. Explain how the area changes when the radius changes.
r = 6 mA = 12π m²
The arc length stays at 4π m. The total perimeter changes.
Work with a partner · Begin without a hint
What could this sector look like?
Total perimeter
34 cm
Area
60 cm²
Find all possible radii and central angles. The sector may be minor or major, but its angle is less than 360°.
Let the arc length be s. Write equations before using your calculator. Show how you know that your answers satisfy both conditions.
P = 2r + s A = ½rs
Calculate · Sketch · Justify
Do both answers make sense?
For P = 34 cm and A = 60 cm², the equation gives r = 5 cm or r = 12 cm.
Radius / cm
Arc length / cm
Angle / degrees
Minor or major?
5
…
…
…
12
…
…
…
s = 34 − 2r θ = 360s2πr
Sketch each sector. Check 0 < s < 2πr, then verify its area and perimeter.
Use your graphing calculator
Two intersections. Two sectors.
With total perimeter fixed at 34 cm:
A(r) = ½r(34 − 2r) A(r) = 17r − r²
Graph y = 17x − x² and y = 60.
What does each intersection represent? Which solution remains if the design must be a minor sector?
Suggested window: 0 ≤ x ≤ 17, 0 ≤ y ≤ 80. Check the geometry after finding the intersections.
20 minutes
Continue with your course route.
SL
A path across a circular park
Combine sector measures with triangle area and the cosine rule.
Find the area and boundary of a circular segment. Hints are available after your first attempt.
HL
A new way to measure angles
Compare arc length with radius, then use radian measure.
Work through the ratio, definition, conversions and application. Write your reasoning.
After your route, go to the individual exit task. On your own device, use the slide menu to return to a question.
SL · Work with a partner
A straight path across a circular park.
The park has radius 12 m. Radii OP and OQ form an angle of 135°. A straight path joins P to Q.
Find the area and total boundary length of the shaded minor segment.
Treat the path as a line with negligible width. Identify the lengths and areas you need. Show exact working, then give answers to 3 significant figures.
The shaded segment is bounded by the arc and the chord PQ.
Finished? Explain why adding 2 × 12 to the arc length does not give the segment perimeter.
SL · Compare after completing your attempt
Check the region and its boundary.
Radius 12 m · central angle 135° · shaded minor segment.
Area
What must be subtracted from the sector?
Boundary length
Which two edges enclose the segment?
Correct any mistake in your working. Explain the correction in one sentence.
HL · Compare with a partner
Same angle. Different circles.
Each sector has central angle 60°. Arc length and radius are measured in centimetres.
Radius r
Arc length s
Ratio s/r
3
π
…
6
2π
…
9
3π
…
Calculate each ratio exactly. Explain why enlargement changes the lengths but not the ratio.
HL · Definition
One radian: arc length equals radius.
The angle is 1 radian when the arc length is equal to the radius.
θ = sr radians
An arc is three radius-lengths long. What is the angle in radians?
Use the same length unit for s and r. Radian measure tells us how many radius-lengths fit along the arc.
HL · Write exact answers
A full turn is 2π radians.
θ = 2πrr = 2π ⟹ 360° = 2π rad
Degrees → radians
Multiply by π180
Convert 90° and 135°.
Radians → degrees
Multiply by 180π
Convert π/3 rad and 5π/6 rad.
Explain why 1 radian is neither 1° nor π radians.
HL · Derive, then check
Use the angle in radians.
Arc length
Rearrange the definition.
θ = sr
Make s the subject.
Sector area
Substitute your result into:
A = ½rs
Write A in terms of r and θ.
Why must θ be in radians in the new formulas?
HL · On your own
Find the radius and angle.
A sector has area 45 cm² and arc length 9 cm.
Find its radius and central angle in radians.
A = ½rs s = rθ A = ½r²θ
Choose a starting formula. Show your working. Check the area using the radius and angle you found.
HL · Interpret without converting to degrees
Which one is the major sector?
Return to the sectors with perimeter 34 cm and area 60 cm².
Radius 5 cm; arc 24 cm
θ = 245 = 4.8 rad
Radius 12 cm; arc 10 cm
θ = 1012 = 56 rad
Compare each angle with π and 2π. Identify the major sector and explain why both are valid.
Everyone · On your own · 10 minutes
Same curved boundary. A new sector.
Sector 1 has radius 9 cm and total perimeter 30 cm.
Sector 1
Find its arc length.
Find its central angle in degrees.
Find its area.
Sector 2
Same arc length as Sector 1; radius 12 cm.
Find its area without first finding its angle. Explain why it is greater.
Does it also have perimeter 30 cm? Justify your answer.
HL: also give both central angles in radians. Show your working.
Review your working
Keep these connections.
Any sector below one full turn
P = 2r + s A = ½rs
Arc length and perimeter are different quantities. Sketch and check every inverse solution.
Angle formulas
When θ is measured in degrees:
s = θ3602πr A = θ360πr²
HL: when θ is measured in radians:
s = rθ A = ½r²θ
Find one answer you corrected today. Write the mathematical reason for the correction.