Area of Trapezoids made easy.

Area of Trapezoids made easy.

In the world of quadrilaterals, the formula to find area of Trapezoids is one of the most feared by middle school students. Perhaps it’s the use of parentheses. If you throw these bad boys into a problem it’s sure to create a certain level of anarchy. Possibly it’s the use of the numbers 1 and 2 written below and to the right of the two bases (b) in the parentheses.

Regardless of of the reason, the formula for Area of Trapezoids has caused many kids to surrender without so much as a fight.

The kicker is that the formula is not difficult to solve. In this blog we will look at a couple of the key terms, dissect the formula, give three examples of finding area and show you a three ways to model finding the area.

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Area of Squares, Rectangles, Rhombuses, & Parallelograms

If you ever google: "How to find the area of a Square? Rectangle? Rhombus? or Parallelogram?" You are going to find a bunch of different answers.

For a Square, the formula given will be the Side Squared. The Rectangle's formula given is Length x Width. To find the area of a Rhombus you are given two different formulas: Base x Height or Diagonal #1 x Diagonal #2 divided by 2.  Finally, the Parallelogram's formula for area is Base x Height.

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All of these formulas are obviously correct, but you can make your life much simpler (or at least your kids) just using one simple formula:

Base x Height.

How is this even possible? The reason it's possible is because all four objects are Parallelograms. For a lot of students, it doesn't make sense how a Square, Rectangle and Rhombus  are also Parallelograms. Below are the rules for an object to qualify as a Parallelogram. 

  • Must have 4 sides

  • All 4 sides must be straight

  • Opposite sides must be the same length

  • Opposite sides must be parallel (so there are two sets of parallel sides)

  • Opposite angles must be the same

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To the right we can see that the Square, Rhombus, and Rectangle each have a total of four sides and each of the sides are straight lines.

The blue and orange highlighted sides are parallel with each other. So each object has a total of two sets of parallel sides.

The green arcs (angles) are opposite of each other. These two angles are the same. In the Square and Rectangle all the angles are 90 degrees, but in the Rhombus the two opposite green angles are both acute angles (less than 90 degrees).

Opposite black arcs (angles) are also the same. In the Rhombus the opposite blacks arcs are each obtuse angles.

Because all 4 shapes (square, rectangle, rhombus & parallelogram) are parallelograms we can just use BASE x HEIGHT to find the area of all four of these shapes.

Modeling Area of Parallelograms

Squares and Rectangles are the two easiest shapes to model. Grid paper is a very easy way to show this to students.

On both of these images not only can you see the base and height but you can visualize the individual squares that make up the rectangle and square. So when we say the area of the rectangle is 10 cm squared we can see 10 squares that have sides that are each 1 cm

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Modeling with Cheez-its

An easy way to engage students is to use Cheez-its (or other square food items like Starburst) when modeling. It's a simple way to turn this into a hands on lesson and the kids like the payoff at the end.

Modeling a Rhombus or Parallelogram (non square/rectangle)

Modeling the area of a Rhombus or parallelogram is a bit harder with cheez-its (cutting up a cheez-its = pure mess). But it is fairly easy to model with grids. We can even turn the grids into a hands on lesson.

Below we have a Parallelogram (base of 8 with a height of 4) and cut it into two pieces by cutting on the height. We can rearrange the two sections to form a rectangle with a base of 8 and a height of 4 (same dimensions as before). By rearranging the Parallelogram we can visualize the 32 squares that make up the area

Below is another example of a parallelogram. We can take this parallelogram with a base of 10 and a height of 3 and cut into two parts by cutting on the height line. The two sections can be rearranged to form a rectangle with a base of 10 and a height of 3. By rearranging the Parallelogram we can visualize the 30 squares that make up the area.

We can model a Rhombus the exact same way by cutting into two parts (along the height) and then rearranging. 

Length x Width vs. Base x Height

Is Length x Height the same as Base x Height?

The answer is Yes (sort of). It all depends on your perspective. On the image below left we are looking down on the rectangle. So, we would refer to these dimensions as the length and the width.

On the image below right we have changed our perspective of the rectangle. We are now looking straight ahead, so our dimensions are now base and height.

So Base and Length and Height and Width are basically the same.

When looking down on a rectangle we can see length and width.

When looking down on a rectangle we can see length and width.

When the same rectangle is upright (like above) we can visualize the height and base.

When the same rectangle is upright (like above) we can visualize the height and base.

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If you are looking for a more challenging activity for you kids check out "Finding Area of Unique Shapes" on Amazon.

This is a collection of 44 different shape combinations using rectangles, triangles, trapezoids, and circles.

The shapes progress in their difficulty.

You can check a video preview of this product below.



Cereal Box Project: A middle school math project

For over a decade the Cereal Box Project has  been a staple in our classroom. The idea is that a cereal box company is introducing a new cereal and they want a unique design for the cereal box to help gain extra attention.

Because the company wants something creative, a simple rectangular prism simply just won't do. The goal is for students to find ways to combine different shapes together to create this eye catching cereal box.

The following is a basic outline of the steps that we follow with this project. In addition, I'm including a few lessons from this years Cereal Box Project experience. 

Step 1 - Come up with an Idea

The idea? A light bulb of course. How do you make a light bulb? Make a Hendecagon of naturally.

The idea? A light bulb of course. How do you make a light bulb? Make a Hendecagon of naturally.

Like baseball? Really into Star Wars? Love doughnuts?

When coming up with an idea I find it helps student engagement when the students can use a topic that they are interested in. For example, if a student is really into baseball/softball they can make their cereal box look like a glove, bat, baseball field, home plate etc. This year a lot of my students are into the video game Fortnite. As a result, a large amount of cereal boxes has a strong Fortnite theme.


Step 2 - Reverse Engineer

As a middle school or elementary student there are certain limitations that we must work around. Many of these limitations are related to curves. For example, If you are trying to make a basketball (sphere), Football (prolate spheroid), or perhaps a megaphone (truncated cone) you are going to dive into a lot of math that is above middle school math level. 

Have the students think about what they want to create and how they can arrange different shapes to make this object. For me, the fun and challenging part of the project is to create these difficult shapes by using the students current knowledge and skills. 

Step 3 - Sketch out the idea

Put it down on paper. Draw the different shapes you are going to combine together to create your shape. Include dimensions.

Step 4 - Build a Prototype

I have our students build their prototype the same size as their actual project. We use scrap paper and masking tape to build these prototypes. A lot of potential pitfalls can be avoided during this step. 

Step 5 - Cutting out individual Pieces

When you begin to build your actual Cereal Box the first thing I like for students to do is cut out the individual sides. We build our cereal boxes out of card stock which is a real thick paper similar to poster board. The thicker paper is nice because it is more stable.


Step 6 - Decorate the individual Pieces

Before the students begin to assemble their projects they should first decorate.  Every year I will have a small collection of students that will tape their entire box together then decide to color and design it.  This almost always leads to frustration and crushed cereal boxes.

Step 7 - Assemble the Cereal Box

We use scotch tape on the final project. We try to tape as much on the inside as possible. As a general rule: The less tape you see the better the project will look.

Step 8 - Complete the required paperwork

We usually have students calculate the surface area and volume of their projects. In addition we also have students draw (to scale) Nets and 3D designs of the project. 

Lessons Learned from this years Cereal Box Projects

We had another successful year with our cereal box project. Many of the projects were creative and well done. Most of the students were fully engaged in the project. But we did have a couple areas that I need to clean up before next year.

The Curse of the Circle

We had a few students this year build cereal boxes that were either cylinders or had a circle as part of the design.  In both of these cases many students just randomly made a circle and just wrapped a long stretch of paper around the circle. There was no thought given to diameter, radius, or circumference. Needless to say, this made for a mess when it came time to make the cereal box the same as its prototype and with calculating surface area and volume.  

Struggles with Scissors and Straight lines

Normally I will take a day (or two) to show the students a few tips/tricks when making and cutting out the sides of their project.  Because of a time crunch this year, I thought I could save a little time and just give a brief talk and turn them lose. In the end this just created a lot more work for me and made it harder to help out as many students.




Interested in giving the Cereal Box Project a try?

This book will give you an in depth guide to how I use this project in my class.