SME
Grade 11 Science · Problem type
Angle in radians and arc length
Use this for “A wheel of radius 0.4 m turns through 135°. What is that angle in radians, and how far does a point on its rim travel?”
Diagram
Sector
Radians
A central angle of 3π/4 radians is 3/8 of a full turn of 2π.
s = r × θ = 0.4 × 3π/4
s = 3π/10 ≈ 0.94 m
Angle in degrees: 135°
This is an example. Type your own values to start a new problem.
θ (rad)
=
θ°
°
×
π
180
s (m)
=
r (m)
×
θ (rad)
Formulas
θ
=
θ
°
×
π
180
3π/4
=
135
×
π
180
s
=
r
×
θ
0.942 = 0.4 × 3π/4
Assumptions
•
One radian is the angle whose arc is one radius long, so a full turn of 360° is 2π radians.
•
To change degrees to radians, multiply by π/180.
•
The arc length is s = rθ, with θ in radians: the radius times the angle.
Step-by-step
WE KNOW
θ° = 135°, r = 0.4 m
FIND
angle in radians (θ), arc length (s)
Step 1 · Find angle in radians (θ)
Use θ = θ° × π/180
Half
a
turn
is
180°,
or
π
radians:
each
degree
is
π
180
radians.
θ
=
135
×
π
180
θ
=
3π/4
rad
(≈
2.36
rad)
Step 2 · Find arc length (s)
Use s = r × θ
Each radian of turn moves a point on the rim one radius along the circle.
s
=
0.4
×
3π/4
s
=
0.942
m
Check
Put the numbers back in. Both sides should match.
3π/4
=
135
×
π
180
✓
0.942 ≈ 0.4 × 3π/4 ✓
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