Justifying Area

Our recent emphasis in the Math Workshop has been with estimating and justifying the area of irregular shapes.

One tool that we have used to justify the area formulas for rectangles and right triangles has been the Geoboard.

Formula for Rectangles (Area = Base x Height)
Formula for Triangles (Area = 1/2 Base x Height)




Consider the irregular polygon shown on the Geoboard template below.

What is the area of this pentagon?  During Math Workshop, we have approached the task of finding solutions to problems similar to this by decomposing the figure into smaller rectangles and right triangles so that we can use the area formulas for the smaller parts and then put those smaller totals together to find a justified total area. 

The green rectangular area in this pentagon has a base of 3 units and a height of 2 units, so the area of this rectangle is 6 square units.   Area of a Rectangle  = Base x Height, so 6 = 3 x 2.

The right scalene triangle (red) shown on the grid is exactly HALF of 2 square units (the dotted lines are shown to illustrate this idea). 

Therefore, the area of the triangle is 1 square unit.
Area of a Triangle = 1/2 Base x Height

The base of the triangle is 1 unit and the height of the triangle is 2 units.
(1 x 2) ÷ 2 = 1

A right scalene triangle is exactly half of a rectangle. 

The right isosceles triangle (blue) shown on the grid is exactly HALF of 1 square unit (the dotted lines are shown to illustrate this idea). 

Therefore, the area of this triangle is 1/2 square unit.

Again, Area of a Triangle = 1/2 Base x Height

The base of this triangle is 1 unit and the height of the triangle is 1 unit.

(1 x 1) ÷ 2 = 1/2

A right isosceles triangle is exactly half of a square.

All decomposed parts are then combined to identify the total area of the figure. 

The area of the pentagon is 7 1/2 square units!!

Students, you can have fun with virtual Geoboards by clicking the link below, which will take you to the Virtual Library of Math Manipulatives. (A parent may have to install Java in order for you to utilize this site.)

Click Here: VIRTUAL GEOBOARD FUN

Also, if you would like to practice with transformations, you can click the links below for some more virtual fun! 

Virtual TRANSLATIONS (Slides)  

Virtual REFLECTIONS (Flips)

Virtual ROTATIONS (Turns)

If you utilize these online resources, leave a comment to let your teachers know so you can earn some extra Behavior Bucks!