Arc Length worksheet

1. Arc length (s): 10.47

1. Half the chord and the radius form a right triangle, so the central angle is θ = 2 arcsin(c ÷ 2r)

   $\theta$$=2\times \text{arcsin}\text{(}\frac{10}{2\times 10}\text{)}$$=60\text{°}$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=10\times 1.047198$$=10.471976$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times {10}^2\times 1.047198$$=52.359878$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 10+10.471976$$=30.471976$

2. Arc length (s): 9.33

1. Work the sector-area formula backwards to find the angle, because the area tells you how wide the slice is

   $\theta$$=\frac{2\times 42}{9^2}$$=1.037037$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=9\times 1.037037$$=9.333333$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times 9^2\times 1.037037$$=42$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 9+9.333333$$=27.333333$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 9\times \text{sin}29.708923\text{°}$$=8.920691$

3. Arc length (s): 11.83

1. Half the chord and the radius form a right triangle, so the central angle is θ = 2 arcsin(c ÷ 2r)

   $\theta$$=2\times \text{arcsin}\text{(}\frac{11}{2\times 9}\text{)}$$=75.339774\text{°}$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=9\times 1.314927$$=11.834344$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times 9^2\times 1.314927$$=53.254548$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 9+11.834344$$=29.834344$

4. Arc length (s): 14.66

1. Arc length needs the angle in radians: multiply degrees by π/180

   $\theta$$=70\times \frac{\pi }{180}$$=1.22173$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=12\times 1.22173$$=14.660766$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times {12}^2\times 1.22173$$=87.964594$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 12+14.660766$$=38.660766$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 12\times \text{sin}35\text{°}$$=13.765834$

5. Arc length (s): 8.64

1. Arc length needs the angle in radians: multiply degrees by π/180

   $\theta$$=55\times \frac{\pi }{180}$$=0.959931$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=9\times 0.959931$$=8.63938$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times 9^2\times 0.959931$$=38.877209$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 9+8.63938$$=26.63938$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 9\times \text{sin}27.5\text{°}$$=8.311475$

6. Arc length (s): 9.77

1. Arc length needs the angle in radians: multiply degrees by π/180

   $\theta$$=70\times \frac{\pi }{180}$$=1.22173$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=8\times 1.22173$$=9.773844$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times 8^2\times 1.22173$$=39.095375$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 8+9.773844$$=25.773844$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 8\times \text{sin}35\text{°}$$=9.177223$

7. Arc length (s): 21.99

1. Arc length needs the angle in radians: multiply degrees by π/180

   $\theta$$=90\times \frac{\pi }{180}$$=1.570796$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=14\times 1.570796$$=21.991149$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times {14}^2\times 1.570796$$=153.93804$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 14+21.991149$$=49.991149$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 14\times \text{sin}45\text{°}$$=19.79899$

8. Arc length (s): 13.61

1. Arc length needs the angle in radians: multiply degrees by π/180

   $\theta$$=65\times \frac{\pi }{180}$$=1.134464$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=12\times 1.134464$$=13.613568$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times {12}^2\times 1.134464$$=81.681409$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 12+13.613568$$=37.613568$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 12\times \text{sin}32.5\text{°}$$=12.895191$

9. Arc length (s): 8.89

1. Work the sector-area formula backwards to find the angle, because the area tells you how wide the slice is

   $\theta$$=\frac{2\times 40}{9^2}$$=0.987654$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=9\times 0.987654$$=8.888889$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times 9^2\times 0.987654$$=40$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 9+8.888889$$=26.888889$

5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

   $c$$=2\times 9\times \text{sin}28.294212\text{°}$$=8.531987$

10. Arc length (s): 7.15

1. Half the chord and the radius form a right triangle, so the central angle is θ = 2 arcsin(c ÷ 2r)

   $\theta$$=2\times \text{arcsin}\text{(}\frac{7}{2\times 10}\text{)}$$=40.97463\text{°}$

2. Arc length is the radius times the angle in radians: s = rθ

   $s$$=10\times 0.715142$$=7.151422$

3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

   $A$$=\frac{1}{2}\times {10}^2\times 0.715142$$=35.75711$

4. The sector's perimeter adds the two radii to the arc

   $P$$=2\times 10+7.151422$$=27.151422$