SME
Grade 10 Math · Problem type
Arc length and sector area in radians
Use this for “A sector has radius 4 cm and central angle 2.5 radians. Find its arc length and area.”
Diagram
Radians
Sector
Each colored arc is one radius long (r = 4 cm).
The angle it makes at the center is 1 radian ≈ 57.3°.
About 6.28 of them go around: a full turn is 2π radians = 360°.
The shaded angle, 2.5 radians, has an arc 2.5 radii long: 10 cm.
This is an example. Type your own values to start a new problem.
s (cm)
=
r (cm)
×
θ
rad
A (cm²)
=
1
2
×
r
2
×
θ
θ°
Central angle in degrees
worked out (not typed) · °
Formulas
s
=
r
×
θ
10 = 4 × 2.5
A
=
r
2
×
θ
2
20
=
4
2
×
2.5
2
θ
°
=
θ
×
180
π
143.2394
=
2.5
×
180
π
Assumptions
•
One radian is the angle whose arc is one radius long; a full turn is 2π radians, or 360°.
•
In radians the arc is s = rθ and the sector’s area is A = ½r²θ.
•
To change radians to degrees, multiply by 180/π.
Step-by-step
WE KNOW
r = 4 cm, θ = 2.5
FIND
arc length (s), sector area (A), central angle in degrees (θ°)
Step 1 · Find arc length (s)
Use s = r × θ
In radians the arc is the radius times the angle.
s
=
4
×
2.5
s
=
10
cm
Step 2 · Find sector area (A)
Use A = r² × θ ÷ 2
The
angle’
s
share
of
πr²:
θ
2π
×
πr²
is
r
2
θ
2
.
A
=
4
2
×
2.5
2
A
=
16
×
2.5
2
A
=
40
2
A
=
20
cm²
Step 3 · Find central angle in degrees (θ°)
Use θ° = θ × 180 ÷ π
π radians is 180°.
θ
°
=
2.5
×
180
π
θ
°
=
450
π
θ
°
=
143.2394°
Check
Put the numbers back in. Both sides should match.
10 = 4 × 2.5 ✓
20
=
4
2
×
2.5
2
✓
143.2394
≈
2.5
×
180
π
✓
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Arc length, sector area and radian measure
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Surface area and volume of solids, cross sections and Cavalieri's principle
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