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Arc length and sector perimeter

Use l = rθ to find arc lengths and sector perimeters. Work with major arcs, chords, annular sectors and curved boundaries using a manual model and original worked questions.

Before you startRadian measure, trigonometric ratios and the cosine rule

01 / Trace the boundary first

An arc length and a sector perimeter are different quantities.

An arc is a curved part of a circle. A sector is enclosed by an arc and two radii. Finding its perimeter means adding all three boundary pieces.

Arc length l = rθ
Sector perimeter P = rθ +2r

θ must be in radians. Here r>0 and the selected sector angle is between 0 and 2π.

Switch between minor and major arcs in the model. The endpoints and radius stay fixed, but the curved distance changes. The chord is shown only for comparison.

Which arc belongs to the boundary?Explore

Radius4. Arc length2π ≈6.283185. Sector perimeter 8 +2π ≈14.283185.

The straight chord has length4√2 ≈5.656854; it is not the arc.

Gold: selected arc. Blue: the two radii to add for the sector perimeter. Dashed green: the chord, which is not part of this sector’s boundary. Lengths use one common unit.

Radius 4Minor angle in steps of π/6 π/2Selected arc

Watch the three boundary pieces separate

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / Why l = rθ

A sector occupies θ/(2π) of a full rotation.

Take the same fraction of the circumferenceWorked example

l = [θ/(2π)] ×2πr =rθ

The 2π factors cancel because θ is measured in radians.

For r =7 cm and θ =1.2 rad, l =8.4 cm

The corresponding sector perimeter is 8.4 +14 =22.4 cm.

If the angle is given in degrees, convert first. For r =6 cm and angle 75°, θ =5π/12 and l =6 ×5π/12 =5π/2 cm. Multiplying 6 by 75 would treat 75 radians as the angle.

03 / Find the missing radius or angle

Rearrange before substituting.

r = l/θ when θ>0
θ = l/r when r>0

Two inverse problemsWorked example

Arc 9 cm with angle 0.75 rad: r =9/0.75 =12 cm

The angle is dimensionless; the answer remains a length.

Arc 5π cm on radius 6 cm: θ =5π/6 rad

Keep π exact.

Arc 48 mm on radius 8 cm: θ =48/80 =0.6 rad

Convert the radius to millimetres first.

Check whether the result matches the stated region: a minor sector has 0<θ<π, a semicircle has θ=π, and a major sector has π<θ<2π.

04 / Use the whole sector perimeter

Both straight sides have length r.

P = r(θ +2)

Find r or θ from the perimeterWorked example

P =30 cm, θ =1 rad: r =30/(1 +2) =10 cm

Do not use 30/1: that would treat the whole perimeter as an arc.

P =35 cm, r =7 cm: 35 =7θ +14

Subtract the two radii before dividing.

θ =3 rad

This is slightly less than π, so the sector is minor.

The formula describes a proper sector with two exposed radii. A complete disc has perimeter 2πr: once the angle is a full turn, the two radii are no longer exterior boundary pieces.

05 / Recover the angle from a chord

Bisect the isosceles triangle, then choose the required arc.

Chord c =2r sin(θ/2)
Minor central angle θ =2arcsin[c/(2r)]

Use0≤c≤2r and the principal inverse sine. Nondegenerate minor arcs have0<θ<π.

A chord 9√2 cm belongs to a circle of radius 9 cmWorked example

sin(θ/2) =(9√2)/(18) =√2/2

Half the minor angle lies between 0 and π/2.

θ/2 =π/4, so θ =π/2

This is the minor angle.

Minor arc =9π/2 cm

The major angle is2π −π/2 =3π/2.

Major arc =27π/2 cm

The two arcs add to the circumference 18π.

The chord is a straight line; neither arc equals the chord. A diameter gives two equal semicircular arcs, so “minor” and “major” no longer distinguish them.

06 / An annular sector has two arcs

Add the outer arc, inner arc and two short straight ends.

P = Rθ +rθ +2(R −r)
= (R +r)θ +2(R −r)

For0<r<R and0<θ<2π. Use the same central angle for both arcs.

Outer radius 7 cm, inner radius 4 cm, angle 1.1 radWorked example

Outer arc 7.7 cm; inner arc 4.4 cm

Both curved edges are exposed.

Each straight end is 7 −4 =3 cm

The straight ends are not full radii.

Perimeter =7.7 +4.4 +3 +3 =18.1 cm

Tracing the outline prevents a missing edge.

A complete annulus has only its two circular boundaries. As with a full disc, do not add imaginary straight cuts when no sector ends are exposed.

07 / Use the angle at the centre

An angle elsewhere on the circle is not the angle in rθ.

A point C lies on the major arc AB. Angle ACB =π/7; radius 14 cm.Worked example

The minor central angle AOB =2π/7

The angle at the centre is twice the angle at the circumference standing on the same minor arc.

Minor arc AB =14 ×2π/7 =4π cm

Use the central angle in l=rθ.

Identify which arc the circumference angle subtends. The point C is on the other arc. A diagram showing a reflex central angle must be interpreted using that same arc, rather than automatically choosing the smaller angle.

08 / A curved edge in a larger shape

Construction lines help calculate, but do not all belong in the perimeter.

Triangle OAB has OA=12 cm, OB=5 cm and ∠AOB=π/3. A circle centred at O with radius 5 meets OA at C. Remove sector OCB; find the perimeter of the remaining shaded region.

Trace A → B → arc BC → AWorked example

AB² =12² +5² −2(12)(5)cos(π/3) =109

Use the cosine rule for the long sloping edge.

AC =12 −5 =7 cm

Subtract the removed radius from OA.

ArcBC =5π/3 cm

Its centre is O and its angle is π/3.

Perimeter =√109 +7 +5π/3 cm

Neither OC nor OB is part of the shaded boundary.

The shaded region has boundary AC, AB and the gold arc BC. The dashed construction radii are not part of its perimeter.

09 / Arc length as distance travelled

Count all the turns when the point keeps moving.

A rotating wheel has radius 8 m and completes one turn in 40 sWorked example

Distance in one turn =16π m

Use the whole circumference.

Average speed =16π/40 =2π/5 m/s

Divide distance by elapsed time.

In km/h: (2π/5) ×3.6 =36π/25 km/h

The conversion factor is 3600/1000.

Why is the perimeter of a full disc of radius r not r(2π+2)?

Hint

Which edges are exposed once the circle is complete?

Worked solution

The two radii are internal construction lines, not exterior edges. Only the circumference contributes:2πr.

11 / Recap

Angles in radians; lengths from the actual boundary.

Arc length is rθ with matching length units.
A proper sector adds two radii to its arc.
A major arc uses 2π minus the minor angle.
Use chord geometry or circle theorems to recover the central angle.
Annular sectors have two arcs and two radial differences.
Trace composite boundaries and omit internal construction lines.
For motion, count the whole travelled arc before dividing by time.

Section 1 of 11 · Trace the boundary first