Worked examples
The question
Question: find the arc length s of a circle with radius 6 and angle 2π/3.
The arc is the curved part of the edge.
We know the radius: r = 6.
r = 6
And the angle: θ = 2π/3.
θ = 2π3
1 Write the rule
The rule for the arc length: s = rθ.
θ must be in radians.
s = rθ
Why?
A full turn is 2π.
Then s is 2π times r: the whole edge of the circle.
A smaller angle gives a part of it.
θ = 2π: s = 2πr, the whole edge
Careful!
The rule only works with the angle in radians.
Ours already is.
Careful: θ in radians, not degrees
2 Put the numbers in
Now put the numbers in: r = 6 and θ = 2π/3.
s = 6 × 2π3
3 Work it out
6 times 2π is 12π.
The bottom stays 3.
s = 12π3
12 and 3 can both be divided by 3.
12 ÷ 3 is 4.
3 ÷ 3 is 1.
12 ÷ 3 = 4
3 ÷ 3 = 1
So s = 4π.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 12.57, to the nearest hundredth.
s = 4π ≈ 12.57
4 Check
Let's check another way.
2π/3 is a part of the full turn, 2π: 1/3 of it.
2π32π = 13
The whole edge is 2π r: 12π.
whole edge: 2π × 6 = 12π
1/3 of 12π is 4π.
The same answer!
So it is right.
13 × 12π = 4π
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
▶ Watch it solved
The question
Question: find the arc length s of a circle with radius 10 cm and angle 60°.
The arc is the curved part of the edge.
We know the radius: r = 10 cm.
r = 10 cm
And the angle: θ = 60°.
θ = 60°
1 Change to radians
The rule needs the angle in radians.
So we change it first.
60° is 60 times one degree.
So we multiply 60 by π/180.
60° = 60 × π180
Top times top: 60π.
The bottom stays 180.
60° = 60π180
60 and 180 can both be divided by 60.
It is the biggest number that goes into both.
60 ÷ 60 is 1.
180 ÷ 60 is 3.
60 ÷ 60 = 1
180 ÷ 60 = 3
So 60° = π/3 radians.
60° = π3
2 Write the rule
The rule for the arc length: s = rθ.
θ must be in radians.
s = rθ
Why?
A full turn is 2π.
Then s is 2π times r: the whole edge of the circle.
A smaller angle gives a part of it.
θ = 2π: s = 2πr, the whole edge
Careful!
θ must be in radians.
10 times 60 gives 600: much longer than the whole edge of the circle!
Careful: not s = 10 × 60 = 600 ✗
3 Put the numbers in
Now put the numbers in: r = 10 and θ = π/3.
s = 10 × π3
4 Work it out
10 times π is 10π.
The bottom stays 3.
s = 10π3
So s = 10π/3 cm.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 10.47 cm, to the nearest hundredth.
s = 10π3 cm ≈ 10.47 cm
5 Check
Let's check another way.
60° is a part of the full turn, 360°: 1/6 of it.
60°360° = 16
The whole edge is 2π r: 20π.
whole edge: 2π × 10 = 20π
1/6 of 20π is 10π/3.
The same answer!
So it is right.
16 × 20π = 10π3
Let's look back.
First: change to radians.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
Answer s = 10π3 cm ≈ 10.47 cm
▶ Watch it solved
The question
Question: find the arc length s of a circle with radius 7 and angle 5π/6.
The arc is the curved part of the edge.
We know the radius: r = 7.
r = 7
And the angle: θ = 5π/6.
θ = 5π6
1 Write the rule
The rule for the arc length: s = rθ.
θ must be in radians.
s = rθ
Why?
A full turn is 2π.
Then s is 2π times r: the whole edge of the circle.
A smaller angle gives a part of it.
θ = 2π: s = 2πr, the whole edge
Careful!
The rule only works with the angle in radians.
Ours already is.
Careful: θ in radians, not degrees
2 Put the numbers in
Now put the numbers in: r = 7 and θ = 5π/6.
s = 7 × 5π6
3 Work it out
7 times 5π is 35π.
The bottom stays 6.
s = 35π6
So s = 35π/6.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 18.33, to the nearest hundredth.
s = 35π6 ≈ 18.33
4 Check
Let's check another way.
5π/6 is a part of the full turn, 2π: 5/12 of it.
5π62π = 512
The whole edge is 2π r: 14π.
whole edge: 2π × 7 = 14π
5/12 of 14π is 35π/6.
The same answer!
So it is right.
512 × 14π = 35π6
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
▶ Watch it solved
