Lesson Video: Area of a Parallelogram Mathematics • 6th Grade

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 “perpendicular” 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 𝐴𝐵𝐶𝐷, where 𝐴𝐵 is equal to 8.3 centimeters.

We are told in the question that the length of the sloping side 𝐴𝐵 is 8.3 centimeters. We need to calculate the area of the parallelogram 𝐴𝐵𝐶𝐷. 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 𝐴𝐵𝐶𝐷 has area 67.76 square centimeters.

If we hadn’t 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 could’ve still solved this problem. The triangle 𝐷𝐶𝐹 is congruent to the triangle 𝐴𝐵𝐸. This means that the rectangle 𝐴𝐷𝐹𝐸 will have the same area as the parallelogram 𝐴𝐵𝐶𝐷. 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, we’re given the area of a parallelogram and need to calculate its base.

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

We need to calculate the length of the base of the parallelogram 𝑋𝐿. 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 𝐿𝑀 is equal to 20.5 centimeters. If we let the length 𝑋𝐿 be 𝑏 centimeters, then the area is equal to 𝑏 multiplied by 20.5. As the area is 610.9, this is equal to 20.5𝑏.

We can calculate the value of 𝑏 by dividing both sides of this equation by 20.5. This gives us 𝑏 is equal to 29.8. The length of 𝑋𝐿 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 𝐴𝐵𝐶𝐷 is equal to 268 square centimeters, find the area of triangle 𝑋𝐵𝐶.

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, we’re 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 𝐵𝐶. They also share the same perpendicular height, the length 𝑋𝑀, 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 𝑋𝐵𝐶 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 𝑋𝐵𝐶 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 𝐴𝐵𝐶𝐷 is 24 square centimeters and the area of the rectangle 𝑋𝐵𝑌𝐷 is 12 square centimeters, find the perimeter of the rectangle 𝑋𝐵𝑌𝐷.

We are given on the diagram that the length of 𝐴𝑋 is three centimeters. We know that the parallelogram 𝐴𝐵𝐶𝐷 has area 24 square centimeters. The area of the rectangle 𝑋𝐵𝑌𝐷 is 12 square centimeters. And we need to calculate the perimeter of this shape. The triangles 𝑋𝐴𝐷 and 𝑌𝐶𝐵 are congruent. This means that they have the same area. This means that we can calculate the area of triangle 𝑋𝐴𝐷 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 𝑋𝐴𝐷 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 ℎ divided by two. Multiplying both sides of this equation by two gives us 12 is equal to three ℎ. We can then divide both sides by three, giving us ℎ is equal to four. The height of the triangle 𝑋𝐷 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 𝑏 multiplied by four. Dividing both sides of this equation by four gives us 𝑏 is equal to three. The base or length of the rectangle 𝑋𝐵 is equal to three centimeters.

We now have a rectangle 𝑋𝐵𝑌𝐷 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 𝑋𝐵𝑌𝐷 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.