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4.5 L) Area of Parallelograms – Part 2
4.5 L) Area of Parallelograms – Part 2
In the previous section, we learnt that we can work out the area of a parallelogram by multiplying the base by the perpendicular height (the perpendicular height is the distance from the base to the opposite side). The formula is given below:
We are now going to have a look at some more complex examples.
Example 1
What is the area of the parallelogram below?
What is the area of the parallelogram below?
In order to use the formula for the area of a parallelogram, we must have the base and the perpendicular height. Currently, we have the base of the parallelogram (15 cm), but we do not have the perpendicular height. However, we are able to find the perpendicular height by using Pythagoras’ theorem. We can use his theorem because we are given a right-angled triangle on the left side of the parallelogram. The unknown length on the right-angled triangle is the perpendicular height of the parallelogram. I have drawn the right-angled triangle below.
Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the square of the other two sides. I am going to let the hypotenuse equal c and the other two sides will be a and b. Here is Pythagoras’ theorem:
We are now able to sub in a as 5 and c as 13.
The next step is to get all of the unknowns to one side of the equation and everything else to the other side. This means that we need to move the 25 from the left to the right, which we do by doing the opposite; we take 25 from both sides of the equation.
We want to find the value of b and not b2. Therefore, we need to square root both sides of the equation.
The perpendicular height of the parallelogram is 12. We can add this value to the diagram.
We are now able to work out what the area of the parallelogram is by subbing the values into the parallelogram formula.
The area of the parallelogram is 180 cm2.
Example 2
What is the area of the parallelogram below? Give your answer to one decimal place.
What is the area of the parallelogram below? Give your answer to one decimal place.
We are given the perpendicular height of the parallelogram (6 cm) and part of the length of the base (part of the base is 5 cm). From the diagram, we can see that the full length of the base will be the length of the base that we are given plus the length of the base for the right-angled triangle on the left of the parallelogram.
For the right-angled triangle, we are given an angle (65°) and the opposite side to the angle. From this information we are able to work out the length that we are looking for, which is the adjacent.
For the right-angled triangle, we are given an angle (65°) and the opposite side to the angle. From this information we are able to work out the length that we are looking for, which is the adjacent.
Therefore, we need to use the trigonometry formula triangle that has an angle, opposite and adjacent. The trigonometry formula triangles are given below.
We are going to be using the TOA formula triangle. We are looking for the adjacent, so we cover up the adjacent and this tells us that we do the following operation:
We are now able to sub the values in.
We now know that the adjacent on the right-angled triangle is 2.7978… cm. It is best to keep the full answer in our calculator for any further calculations rather than rounding. This is because rounding early on in your calculation can result in the final answer being slightly out.
We are now able to find the total length of the base by adding 2.7978… on to the length of the base that we are given (5 cm). Therefore, the base for this parallelogram is 7.7978… cm.
We can now work out the area of the parallelogram.
We are now able to find the total length of the base by adding 2.7978… on to the length of the base that we are given (5 cm). Therefore, the base for this parallelogram is 7.7978… cm.
We can now work out the area of the parallelogram.
The final step is to round the answer to one decimal place.
The area of the parallelogram is 46.8 cm2.