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Word Problems: Area and Perimeter of Parallelograms
In order to solve problems which require application of the area and perimeter for parallelograms, it is necessary to
 
 
A typical problem involving the area and perimeter of a parallelogram gives us the area, perimeter and/or base, height, and an angle of the parallelogram. We may also be given a relationship between the area and perimeter or between the base and height of the parallelogram. We need to calculate some of these quantities given information about the others. Two examples of this type of problem follow:
 
  1. Suppose in a parallelogram the base is 8 and the height is 4. What are the area and perimeter of this parallelogram? A diagram is shown below.
 
 
Notice that s1 > s2. These parallelograms show two of the infinitely many possible parallelograms with a base of 8 and a height of 4.
 
We can find the area of these parallelograms by using A = bh = (8)(4) = 32. We can NOT find the perimeter because there are infinitely many possible parallelograms that can drawn having different lengths for the other two sides.
Notice the importance of making a diagram (or more than one) to see what is happening when using the given information.
 
  1. Suppose a parallelogram has a base of 8, a height of 4, and the side other than the base makes a 41° angle with the base. What are the area and perimeter? A diagram is shown below.
 
 
Notice that the side s is the hypotenuse of a right triangle with an angle of 41° opposite the height which is 4. We can use this relationship to find the length of side s as follows:
 
 
This allows us to find the perimeter which is the sum of the four sides, two bases of length 8 and two sides of length 6.097.
 
The perimeter is 28.194
 
The area is easy to find since we have the base and height.
 
Area = bh = (8)(4) = 32
 
Once again, a diagram is helpful because it clearly showed the right triangle which allowed us to find the length of the side s.

Let's Practice

Question #1
AudioThe base of a parallelogram is 15 and the height is 10. What are the area and perimeter of this parallelogram?


Question #2
AudioA parallelogram has a base of 14, a height of 9, and the base makes an angle of 70° with the other side. What are the area and perimeter?


Question #3
AudioA parallelogram has a base of 3x, a height of x, and the other side of the parallelogram (not the base) is 2x. If the area of this parallelogram is 15, what is its perimeter?



Try These
Question #1
AudioThe base of a parallelogram is 20 and the area is 300. What are the height and perimeter of this parallelogram?


Question #2
AudioA parallelogram has a base of length 30. Another side of length 20 forms a 58° angle with this base. What are the area and perimeter of this parallelogram correct to one decimal place?


Question #3
Audio
A parallelogram is shown in the diagram below. If the perimeter is 38 and the area is 60, what is the height of this parallelogram?
 



This type of problem involves relationships among the lengths of the sides and height as well as the area and perimeter of a parallelogram. We need to focus on two formulas: (1) area which requires a measurement of the base and height, which we call h and (2) the perimeter which is the sum of all four sides. Sometimes it is useful to remember that the opposite sides of a parallelogram are always equal. Sometimes an angle of the parallelogram is also given so that we can use one of the basic trigonometric ratios of sine, cosine, or tangent to calculate the length of the height.
 
Sometimes there is not enough information given to find the perimeter. While the base and height do determine the area, there are infinitely many parallelograms with the same base and height, but different perimeters.

M Ransom

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