Area of a Sector | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Use the fraction of the circle

Let $A$ be the sector area, $r$ its radius and $\theta$ its angle in radians. A full circle has angle $2\pi$ and area $\pi r^2$ , so:

$A=\frac{\theta}{2\pi}\times\pi r^2=\frac12r^2\theta$

The factor $\pi$ cancels; the 2 stays in the denominator. That is why the formula has a half. The formula is supplied in MF19, the exam formula booklet, along with $s=r\theta$ .

Swipe left or right to see the whole diagram →

The sector has two straight radii and one curved arc.

Worked example

Radius 9 cm, angle π/6 radians: find the exact area

$A=\frac12(9^2)\frac{\pi}{6}=\frac{81\pi}{12}$ .

Answer

A=\frac{27\pi}{4}\text{ cm}^2

With the angle fixed, doubling the radius multiplies area by four because the radius is squared. With the radius fixed, doubling the angle doubles the area.

When the radius or angle is missing

To find an angle, divide by $r^2/2$ : $\theta=2A/r^2$ . To find a radius, first obtain $r^2=2A/\theta$ , then take the positive square root because a radius is a length.

For example, area 18 cm² and angle 1 rad give $r^2=36$ , so $r=6$ cm, not −6 cm.

Worked example

Arc length 9.6 cm and sector area 76.8 cm²: find r and θ

(9709/13 May/Jun 2023 Q6(a)) Both are unknown, but both measurements refer to the same sector. Replace $r\theta$ in its area formula by the known arc length:

$A=\frac12r(r\theta)=\frac12rs$ .

$76.8=\frac12r(9.6)=4.8r$ , so $r=16$ cm.
Return to the arc equation: $\theta=9.6/16=0.6$ rad.

Check both measurements: $16(0.6)=9.6$ and $\frac12(16^2)(0.6)=76.8$ .

Use the information you have

Your turn

A sector has arc length 10 cm and area 40 cm². Find its radius and angle. If the radius then doubles while the angle stays fixed, find the new area.

Show worked answer

$40=\frac12r(10)$ gives $r=8$ cm. Then $\theta=10/8=1.25$ rad. The new area is $4(40)=160$ cm².

Use the fraction of the circle

Let $A$ be the sector area, $r$ its radius and $\theta$ its angle in radians. A full circle has angle $2\pi$ and area $\pi r^2$ , so:

$A=\frac{\theta}{2\pi}\times\pi r^2=\frac12r^2\theta$

The factor $\pi$ cancels; the 2 stays in the denominator. That is why the formula has a half. The formula is supplied in MF19, the exam formula booklet, along with $s=r\theta$ .

Swipe left or right to see the whole diagram →

The sector has two straight radii and one curved arc.

Worked example

Radius 9 cm, angle π/6 radians: find the exact area

$A=\frac12(9^2)\frac{\pi}{6}=\frac{81\pi}{12}$ .

Answer

A=\frac{27\pi}{4}\text{ cm}^2

With the angle fixed, doubling the radius multiplies area by four because the radius is squared. With the radius fixed, doubling the angle doubles the area.

When the radius or angle is missing

To find an angle, divide by $r^2/2$ : $\theta=2A/r^2$ . To find a radius, first obtain $r^2=2A/\theta$ , then take the positive square root because a radius is a length.

For example, area 18 cm² and angle 1 rad give $r^2=36$ , so $r=6$ cm, not −6 cm.

Worked example

Arc length 9.6 cm and sector area 76.8 cm²: find r and θ

(9709/13 May/Jun 2023 Q6(a)) Both are unknown, but both measurements refer to the same sector. Replace $r\theta$ in its area formula by the known arc length:

$A=\frac12r(r\theta)=\frac12rs$ .

$76.8=\frac12r(9.6)=4.8r$ , so $r=16$ cm.
Return to the arc equation: $\theta=9.6/16=0.6$ rad.

Check both measurements: $16(0.6)=9.6$ and $\frac12(16^2)(0.6)=76.8$ .

Use the information you have

Your turn

A sector has arc length 10 cm and area 40 cm². Find its radius and angle. If the radius then doubles while the angle stays fixed, find the new area.

Show worked answer

$40=\frac12r(10)$ gives $r=8$ cm. Then $\theta=10/8=1.25$ rad. The new area is $4(40)=160$ cm².