Length of an Arc | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

One radian gives one radius of arc

Since $\theta=s/r$ , multiplying both sides by $r$ gives $s=r\theta$ . Here $r$ is the radius, $s$ is the arc length and $\theta$ must be in radians.

A is the region between two radii and their arc.

Swipe left or right to see the whole diagram →

The sector has two straight radii and one curved arc.

$\angle AOB$ means the angle at O: the middle letter names the point where the two lines meet.

Worked example

Radius 15 cm, angle π/3 radians

$s=15\times\frac{\pi}{3}=5\pi\text{ cm}$ . This is an exact length; its decimal value is approximately 15.7 cm.

If the unknown is different, use the same equation. For arc 12 cm and angle 1.5 rad, $12=1.5r$ , so divide by 1.5 to get $r=8$ cm. For arc 5 cm and radius 10 cm, $\theta=5/10=0.5$ rad.

The perimeter includes the straight sides

means the total boundary length. A sector has an arc and two radii, so $P=s+2r$ .

Worked example

Find the perimeter: radius 10 cm, angle 72°

Convert the angle: $\theta=72\pi/180=2\pi/5$ rad.
Find the arc: $s=10(2\pi/5)=4\pi$ cm.
Add both radii: $P=4\pi+20$ cm.

Answer

P\approx32.6\text{ cm}

are the shorter and longer arcs between the same endpoints. If the minor angle is $\theta$ , the major angle is $2\pi-\theta$ . The same distinction applies to their sectors.

For example, with radius 3 cm and minor angle 1 rad, the major arc is $3(2\pi-1)$ cm. Its sector perimeter adds another 6 cm. Use the angle of the region requested, not automatically the smaller angle shown.

Common mistake

P=r\theta+2r

is for a sector with two distinct bounding radii,

0<\theta<2\pi

. A whole circle has perimeter

2\pi r

; there are no straight edges to add.

Choose the required boundary

Your turn

A circle has radius 5 cm. Its minor arc AB is 4 cm long. Find the central angle and the perimeter of the major sector AOB.

Show worked answer

$\theta=4/5=0.8$ rad. The major arc is the circumference minus 4 cm, so $P=10\pi-4+10=10\pi+6$ cm, approximately 37.4 cm.

One radian gives one radius of arc

Since $\theta=s/r$ , multiplying both sides by $r$ gives $s=r\theta$ . Here $r$ is the radius, $s$ is the arc length and $\theta$ must be in radians.

A is the region between two radii and their arc.

Swipe left or right to see the whole diagram →

The sector has two straight radii and one curved arc.

$\angle AOB$ means the angle at O: the middle letter names the point where the two lines meet.

Worked example

Radius 15 cm, angle π/3 radians

$s=15\times\frac{\pi}{3}=5\pi\text{ cm}$ . This is an exact length; its decimal value is approximately 15.7 cm.

If the unknown is different, use the same equation. For arc 12 cm and angle 1.5 rad, $12=1.5r$ , so divide by 1.5 to get $r=8$ cm. For arc 5 cm and radius 10 cm, $\theta=5/10=0.5$ rad.

The perimeter includes the straight sides

means the total boundary length. A sector has an arc and two radii, so $P=s+2r$ .

Worked example

Find the perimeter: radius 10 cm, angle 72°

Convert the angle: $\theta=72\pi/180=2\pi/5$ rad.
Find the arc: $s=10(2\pi/5)=4\pi$ cm.
Add both radii: $P=4\pi+20$ cm.

Answer

P\approx32.6\text{ cm}

are the shorter and longer arcs between the same endpoints. If the minor angle is $\theta$ , the major angle is $2\pi-\theta$ . The same distinction applies to their sectors.

Common mistake

P=r\theta+2r

is for a sector with two distinct bounding radii,

0<\theta<2\pi

. A whole circle has perimeter

2\pi r

; there are no straight edges to add.

Choose the required boundary

Your turn

A circle has radius 5 cm. Its minor arc AB is 4 cm long. Find the central angle and the perimeter of the major sector AOB.

Show worked answer

$\theta=4/5=0.8$ rad. The major arc is the circumference minus 4 cm, so $P=10\pi-4+10=10\pi+6$ cm, approximately 37.4 cm.