Area of Complex Shapes

One of my classes has been exploring area lately. They started by counting squares and differentiating between what is considered area and perimeter. Students were able to add halves and reasonably estimate what the area of a rectangle, parallelogram and triangle would be based on a brief observation. Becoming precise was not valued early on in the process but proved to be a tough transition as students were expected to use formulas later in the unit. Late last week students were asked to find the area of the shape below.

At first students were fairly confident in being able to find the area. They quickly counted up the squares that were fully visible. Then added the halves or what they perceived as half.

Students knew that there were at least 15 full squares covered and then added the halves. Estimates were given based on the full squares visible and ranged from 20 to 45. Confidence waned during this time as some students erased the numbers and started to deconstruct the shape into smaller shapes.

Earlier in the unit students made the connection that the area of a triangle can be found by using a rectangle method. Students also explored how parallelograms can be modified and rearranged into a rectangle.

Using that understanding, a number of students tried different methods to find the area of the shape. Students worked in groups to find a common understanding of where to start and how to dismantle the shape into parallelograms, triangles and rectangles.

This group decided to split apart the shape into triangles and rectangles. They specifically used the rectangle method to find the area of the triangles and counted the middle.

Another group tried a hybrid approach with mostly triangles and two parallelograms. The problem that this group had was trying to decide what constitutes the base and height of each triangle.

The other group decided to split one side of the shape into triangles and the other side into parallelograms. When I showed this to the class I received a few shocked looks. They were amazed at how simple this looked and yet they came up with the correct answer.

Overall, this was a time consuming task, but I feel like it was worthwhile. Students were able to think about math and measurement a bit differently. There are more efficient ways, but not one right way to complete the task. I am hoping that students remember this task and build upon their understanding as we move towards additional measurement concepts next school year.

Author: Matt Coaty

I've taught elementary students for the past 14 years. I enjoy reading educational research and learning from my PLN. Words on this blog are my own.

One thought on “Area of Complex Shapes”

  1. This is an excellent exercise for your students. Adding and subtracting areas is a frequently tested skill on math contests. You can tell your students that high school level math contests, ones that determine who will represent the US in the International Math Olympiad include many problems like these. The most recent AMC 10 asks students to find the areas of 4 weird looking polygons, including one that vaguely resembles a Star Trek insignia.


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