### Video Transcript

In the diagram, we have a
parallelogram π΄π΅πΆπ· with a sector centred on π΄. The length of the base of the
parallelogram is 12 centimetres and its perpendicular height is eight
centimetres. The radius of the sector is four
centimetres and its angle is 60 degrees. Find the area of the shaded
region. Give your answer to three
significant figures.

In order to find the area of the
shaded region, weβre going to need to perform two separate calculations. First, weβll need to find the area
of the parallelogram, which forms the majority of the shaded area. Next, weβll need to find the area
of the sector and subtract it from the area of the parallelogram.

The formula for area of a
parallelogram is the length of the base multiplied by its perpendicular height. In this case, the length of the
base is 12 centimetres and the perpendicular height of the parallelogram is eight
centimetres. 12 multiplied by eight is 96. The area of the parallelogram is 96
centimetres squared.

Next, we need to find the area of
the sector. Sector area is given by π divided
by 360 multiplied by ππ squared, where π is the given angle. This sector has a radius of four
centimetres and an angle of 60 degrees. So this formula becomes 60 divided
by 360 multiplied by π multiplied by four squared.

Now this sort of question will
usually be on a calculator paper. So we can put these numbers into
the calculator and save ourselves some time. However, it is useful to know how
to simplify this expression just in case youβre asked to give your answer in terms
of π.

First, notice that both 60 and 360
have a highest common factor of 60. 60 divided by 60 is one and 360
divided by 60 is six. This becomes a sixth multiplied by
π multiplied by 16. Next, we notice that 16 and six
have a common factor of two. 16 divided by two is eight and six
divided by two is three. The area of the sector, therefore,
becomes eight π over three centimetres squared.

Remember, we said that, to find the
shaded area, we would take the area of the parallelogram and subtract the area of
the sector. The shaded region is, therefore, 96
minus eight π over three. Thatβs 87.622, which, correct to
three significant figures, is 87.6 centimetres squared.