SME
Grade 10 Math · Geometry
Arc length, sector area and radian measure
Use this for “A circle has radius 6 cm. Find the arc length and the sector area for a 150° central angle.”
Diagram
A central angle of 150° is 5/12 of the full 360°, or 5π/6 ≈ 2.62 radians.
s
=
150
360
×
2π
×
6
=
5π
≈
15.71
cm
A
=
150
360
×
π
×
6
2
=
15π
≈
47.12
cm²
This is an example. Type your own values to start a new problem.
s (cm)
=
θ
360
×
2π
×
r (cm)
A (cm²)
=
θ
360
×
π
×
r
2
Formulas
s
=
θ
360
×
2
×
π
×
r
5π
=
150
360
×
2
×
π
×
6
A
=
θ
360
×
π
×
r
2
15π
=
150
360
×
π
×
6
2
Assumptions
•
A central angle of θ° cuts off θ/360 of the circle.
•
The arc is that share of the circumference 2πr; the sector is that share of the area πr².
•
An angle over 180° gives a major arc and a sector bigger than half the circle.
Step-by-step
WE KNOW
r = 6 cm, θ = 150°
FIND
arc length (s), sector area (A)
Step 1 · Find arc length (s)
Use s = θ ÷ 360 × 2 × π × r
The
arc
is
the
angle’
s
share
of
the
whole
circumference.
s
=
150
360
×
2
×
π
×
6
s
=
5
12
×
12π
s
=
5π
cm
(≈
15.71
cm)
Step 2 · Find sector area (A)
Use A = θ ÷ 360 × π × r²
The
sector
is
the
angle’
s
share
of
the
whole
circle’
s
area.
A
=
150
360
×
π
×
6
2
A
=
5
12
×
36π
A
=
15π
cm²
(≈
47.12
cm²)
Check
Put the numbers back in. Both sides should match.
5π
=
150
360
×
2
×
π
×
6
✓
15π
=
150
360
×
π
×
6
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.