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It is possible to find a formula for the area of geoboard polygons as a function of boundary dots and interior dots. For example, the two polygons below each have five boundary dots and three interior dots:
a. | What is the area of each polygon?
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Let's gather data to help us find what's known as Pick's formula, which is used for determining the area of a simple closed curve (in our case, the areas of the polygons on a geoboard).
For Problems (b)-(d), build figures on the geoboard or draw figures on dot paper that have the indicated number of boundary dots (b) and interior dots (I). Find the area of each figure, and record your results in the tables below.
b. | If I = 0, calculate the area of each figure: |
c. | If I = 1, calculate the area of each figure: |
d. | If I = 2, calculate the area of each figure: |
e. | What patterns do you notice in these tables? Each time you add a boundary dot, how does it change the area? |
f. | Find a formula for the area of a geoboard figure if it has b boundary dots and I interior dots. |
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, where b1 and b2 represent the top and bottom base of the trapezoid and h represents the height. Draw a trapezoid on a sheet of paper, and connect either of the two opposite vertices. Into what shapes has the trapezoid been divided? What are the height and base of each shape? Find the area of each shape and add them together. How does this area compare to the total area of the trapezoid? 
