Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The opposite sides of a parallelogram must be congruent, its opposite angles must also be congruent, and the consecutive angles must be supplementary. Here are a couple of examples illustrating what parallelograms look like:

Parallelograms might look a little funny, but this article will help you work with them.

Finding the area of parallelograms

The formula for the area of a parallelogram is the same as other quadrilaterals such as rectangles:

$A=bh$

In this formula, where A is the area, b is the length of one base, and h is the height. The height can be difficult to figure out since a parallelogram isn't straight, but you can get it by drawing a right angle connecting the base to the top. Here are a couple of examples of what that looks like:

You can also split parallelograms into two congruent triangles with a diagonal line, allowing you to use concepts such as the Pythagorean Theorem or 30-60-90 triangles to calculate the sides and area of the two triangles, which you can then sum to find the total area.

Once you have the base and height, finding the area is as easy as multiplying the two numbers. For example, let's say your parallelogram has a base measuring 7 meters and a height of 4 meters. We would calculate the area as $7\times 4=28\mathrm{sq.\; meters}$ . Don't forget to express your answer in square units!

Finding the perimeter of parallelograms

The perimeter of a parallelogram is the sum of the lengths of its four sides, again just like a rectangle. Since we know that the opposite sides of a parallelogram are congruent, we can find the perimeter even if all four measurements aren't given to us. Consider the following diagram:

We might only know the lengths of two of the sides, but that's all we need assuming figure PQRS is a parallelogram. Opposite sides must be congruent, which means that PQ must measure 10 cm since SR does. Likewise, PS must measure 6 cm since QR does. That means we have values for all four sides, allowing us to calculate the perimeter:

$10+10+6+6=32cm$

Don't forget to include the unit of measurement!

Parallelograms practice questions

a. What is the area of a parallelogram with a base of 6 inches and a height of 8 inches?

To find the area of a parallelogram, use the formula: $\mathrm{Area}=\mathrm{base}\times \mathrm{height}$

$\mathrm{Area}=6\mathrm{ in}\times 8\mathrm{ in}=48\mathrm{ sq.\; in.}$

b. What is the area of a parallelogram with a base of 4 meters and a height of 10 meters?

$\mathrm{Area}=\mathrm{base}\times \mathrm{height}$

$\mathrm{Area}=4\mathrm{ m}\times 10\mathrm{ m}=40\mathrm{ sq.\; m.}$

c. If figure WXYZ is a parallelogram with one side measuring 5 feet and another measuring 9 feet, what is its perimeter?

In a parallelogram, opposite sides are equal. So, we have two sides of 5 ft and two sides of 9 ft.

$\mathrm{Perimeter}=2\left(\mathrm{side1}\right)+2\left(\mathrm{side2}\right)=2\left(5\mathrm{ft}\right)+2\left(9\mathrm{ft}\right)=10\mathrm{ft}+18\mathrm{ft}=28\mathrm{ft}$

d. What is the perimeter of a parallelogram with sides of 8m and 12m?

$\mathrm{Perimeter}=2\left(\mathrm{side1}\right)+2\left(\mathrm{side2}\right)=2\left(8m\right)+2\left(12m\right)=16m+24m=40m$

Topics related to the Parallelogram

Parallel Lines

Perimeter, Area and Volume

Graphing Polygons

Flashcards covering the Parallelogram

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Parallelogram

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

Get help with parallelograms today with Varsity Tutors

Parallelograms are a deceptively challenging topic for many students because they share similarities with rectangles but are often much more abstract in practice. If your student doesn't see how a parallelogram can be divided into triangles or understand how to fill in the missing sides for a perimeter problem, they may experience further difficulties working with more advanced concepts. Luckily, working with an experienced math tutor can help students of all ages and ability levels gain self-confidence and deepen their understanding of quantitative concepts. Contact Varsity Tutors right now to get connected with a qualified tutor.