SME
Grade 12 Math · Problem type
The spiral r = aθ
Use this for “Graph r = 0.5θ: find r at θ = 270°” or “At what angle is r = 6?”
Diagram
r = 0.5θ, θ from 0 to 4π.
(r, θ) = (2.36, 3π/2)
This is an example. Type your own values to start a new problem.
a
Growth a
example
θ
Angle
example · °
r
Distance r
calculated
G
Growth per turn
worked out (not typed)
Formulas
r
=
a
×
θ
×
π
180
2.3562
=
0.5
×
270
×
π
180
G
=
2π
×
a
3.1416 = 2π × 0.5
Assumptions
•
In r = aθ, θ is in radians: θ° × π/180, so r grows by 2πa every turn.
•
With a > 0 the spiral winds out counterclockwise from the pole as θ grows.
•
Two turns are drawn, θ from 0° to 720°.
Step-by-step
WE KNOW
a = 0.5, θ = 270°
FIND
distance r (r), growth per turn (G)
Step 1 · Find distance r (r)
Use r = a × θ × π/180
Change
θ
to
radians,
then
multiply
by
a
.
r
=
0.5
×
270
×
π
180
θ
=
270
×
π
180
=
3π/2
r
=
0.5
×
3π/2
=
3π/4
r
=
2.3562
Step 2 · Find growth per turn (G)
Use G = 2π × a
One
turn
adds
2π
to
θ
,
so
r
grows
by
2π
×
a
:
the
rings
are
evenly
spaced.
G
=
2π
×
0.5
G
=
3.1416
Check
Put the numbers back in. Both sides should match.
2.3562
≈
0.5
×
270
×
π
180
✓
3.1416 ≈ 2π × 0.5 ✓
Topic check
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