### Video Transcript

In this video, we will learn how to
calculate the area of a parallelogram and solve word problems requiring the areas of
parallelogram-shaped figures. We will begin by defining what we
mean by a parallelogram and look at how we can calculate its area.

A parallelogram is a four-sided
shape with two pairs of parallel sides. We can calculate the area of any
parallelogram by multiplying the base by the perpendicular height. The word βperpendicularβ means at
90 degrees. Therefore, the height must be at
right angles to the base. This is the same formula that we
use when calculating the area of a rectangle.

If we cut off the right-angled
triangle at the right-hand side of our parallelogram and added it to the other side,
we would create a rectangle. This rectangle would have the same
dimensions, base and perpendicular height, as the original parallelogram. When calculating the area of a
parallelogram, it is important that we use the perpendicular height and not the
sloping height.

We will now look at some questions
that involve calculating the area of a parallelogram.

Find the area of the parallelogram
π΄π΅πΆπ·, where π΄π΅ is equal to 8.3 centimeters.

We are told in the question that
the length of the sloping side π΄π΅ is 8.3 centimeters. We need to calculate the area of
the parallelogram π΄π΅πΆπ·. We recall that the area of any
parallelogram is equal to its base multiplied by the perpendicular height. In this question, the base of the
parallelogram is equal to 8.8 centimeters and the perpendicular height is 7.7
centimeters. It is important to note that it is
this value we use and not the sloping height, 8.3 centimeters.

To calculate the area, we therefore
multiply 8.8 by 7.7. This is equal to 67.76. As the dimensions of the
parallelogram were in centimeters, the units for the area will be square
centimeters. The parallelogram π΄π΅πΆπ· has area
67.76 square centimeters.

If we hadnβt remembered our formula
for the area of a parallelogram but we had recalled the area of a rectangle is equal
to the base multiplied by the height, we couldβve still solved this problem. The triangle π·πΆπΉ is congruent to
the triangle π΄π΅πΈ. This means that the rectangle
π΄π·πΉπΈ will have the same area as the parallelogram π΄π΅πΆπ·. The rectangle has dimensions 8.8
centimeters and 7.7 centimeters. Multiplying these would give us the
area of the rectangle, which is the same calculation we performed to work out the
area of the parallelogram.

In our next question, weβre given
the area of a parallelogram and need to calculate its base.

Given that the area of the
parallelogram ππππΏ is equal to 610.9 square centimeters, find the length of
ππΏ.

We need to calculate the length of
the base of the parallelogram ππΏ. We are also told that the area of
the parallelogram is 610.9 square centimeters. We recall that the area of any
parallelogram can be calculated by multiplying its base by its perpendicular
height. The perpendicular height πΏπ is
equal to 20.5 centimeters. If we let the length ππΏ be π
centimeters, then the area is equal to π multiplied by 20.5. As the area is 610.9, this is equal
to 20.5π.

We can calculate the value of π by
dividing both sides of this equation by 20.5. This gives us π is equal to
29.8. The length of ππΏ is therefore
equal to 29.8 centimeters.

In our next question, we need to
find the area of a triangle drawn inside a parallelogram.

Given that the area of the
parallelogram π΄π΅πΆπ· is equal to 268 square centimeters, find the area of triangle
ππ΅πΆ.

We are told in the question that
the area of the parallelogram is 268 square centimeters. We recall that the area of a
parallelogram is equal to the base multiplied by the perpendicular height. In this question, however, weβre
not given either of these dimensions. We do know, however, that the area
of any triangle is equal to the base multiplied by its height divided by two. Once again, this height must be the
perpendicular height.

In the figure drawn, the
parallelogram and triangle share a base, the length π΅πΆ. They also share the same
perpendicular height, the length ππ, as shown on the diagram. As the area of the parallelogram is
base multiplied by height and the area of the triangle is base multiplied by height
divided by two, the triangle must have half the area of the parallelogram. The area of triangle ππ΅πΆ is
therefore equal to 268 divided by two or half of 268. This is equal to 134. We can therefore conclude that the
area of triangle ππ΅πΆ is equal to 134 square centimeters.

Our next question involves a
parallelogram inside a rectangle.

The given figure shows a
parallelogram inside a rectangle. Determine the area inside the
rectangle that is not occupied by the parallelogram.

In order to answer this question,
we recall that the area of any rectangle is equal to its base multiplied by its
height. The area of a parallelogram is also
equal to its base multiplied by its height. This must be the perpendicular
height and not the slant height. The base of the rectangle is 72
centimeters. We need to add 42 and 30. This means that the area is equal
to 72 multiplied by 28. This is equal to 2016. As the dimensions were in
centimeters, the area of the rectangle is 2016 square centimeters.

The parallelogram has a base of 42
centimeters. It has the same height as the
rectangle, 28 centimeters. This means that the area is equal
to 42 multiplied by 28. This is equal to 1176. Therefore, the area of the
parallelogram is equal to 1176 square centimeters.

We need to calculate the area
inside the rectangle that is not occupied by the parallelogram. To calculate this area, we need to
subtract 1176 from 2016. This is equal to 840. The area that is inside the
rectangle that is not occupied by the parallelogram is 840 square centimeters.

An alternative method here would be
to consider the two right-angled triangles. These two triangles are congruent,
so we could fit them together to make a rectangle. This rectangle has a base of 30
centimeters and a height of 28 centimeters. Therefore, its area is equal to 30
multiplied by 28. Once again, this gives us an answer
of 840 square centimeters.

Our final question is a more
complicated problem where the rectangle is inside the parallelogram.

Given that the area of the
parallelogram π΄π΅πΆπ· is 24 square centimeters and the area of the rectangle
ππ΅ππ· is 12 square centimeters, find the perimeter of the rectangle ππ΅ππ·.

We are given on the diagram that
the length of π΄π is three centimeters. We know that the parallelogram
π΄π΅πΆπ· has area 24 square centimeters. The area of the rectangle ππ΅ππ·
is 12 square centimeters. And we need to calculate the
perimeter of this shape. The triangles ππ΄π· and ππΆπ΅ are
congruent. This means that they have the same
area. This means that we can calculate
the area of triangle ππ΄π· by subtracting 12 from 24 and then dividing by two.

Subtracting the area of the
rectangle from the area of the parallelogram will give the area of both
triangles. As the triangles are congruent, we
then need to divide by two. 24 minus 12 divided by two is equal
to six. The area of triangle ππ΄π· is six
square centimeters.

We know that to calculate the area
of any triangle, we multiply the base by the height and then divide by two. We already know that the base of
this triangle is three centimeters. This means that six is equal to
three multiplied by β divided by two. Multiplying both sides of this
equation by two gives us 12 is equal to three β. We can then divide both sides by
three, giving us β is equal to four. The height of the triangle ππ· is
equal to four centimeters.

We know the area of any rectangle
is equal to its base multiplied by its height. We know the height of the rectangle
is four centimeters and its area is 12 square centimeters. Substituting in these values gives
us 12 is equal to π multiplied by four. Dividing both sides of this
equation by four gives us π is equal to three. The base or length of the rectangle
ππ΅ is equal to three centimeters.

We now have a rectangle ππ΅ππ·
with dimensions four centimeters and three centimeters. The opposite sides of a rectangle
are equal in length, and the perimeter is the distance around the outside. We can therefore calculate the
perimeter by adding two threes and two fours. This is equal to 14. The perimeter of rectangle ππ΅ππ·
is 14 centimeters.

We will now summarize the key
points from this video. We can calculate the area of a
parallelogram by multiplying its base by its perpendicular height. This is the same formula we use to
calculate the area of a rectangle, as the two triangles shown are congruent. We also saw how we could solve
problems involving triangles, rectangles, and parallelograms inscribed within each
other.