Learning Math: Measurement
Area Homework
Session 6, Homework
Problem H1
If the area of each of the smallest two triangles in the tangram below is equal to 1 unit, find the area of each of the other pieces, and then find the area of the entire tangram:
Problem H2
The formula for the area of a trapezoid is A = , where b_{1} and b_{2} 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?
Problem H3
Examine the trapezoid below:
Find the area of this trapezoid using the formula A = . Then find the area in a different way. How do the two areas compare?
Take It Further:
Problem H4
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:
 What is the area of each polygon?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.
 If I = 0, calculate the area of each figure:
Number of Boundary Dots
Area (in Square Units)
3 4 5 6 7 b  If I = 1, calculate the area of each figure:
Number of Boundary Dots
Area (in Square Units)
3 4 5 6 7 b  If I = 2, calculate the area of each figure:
Number of Boundary Dots
Area (in Square Units)
3 4 5 6 7 b  What patterns do you notice in these tables? Each time you add a boundary dot, how does it change the area?
 Find a formula for the area of a geoboard figure if it has b boundary dots and I interior dots.
Suggested Reading
Fan, C. Kenneth (January, 1997). Areas and Brownies. Mathematics Teaching in the Middle School, 2 (3), 148160.
Reproduced with permission from Mathematics Teaching in the Middle School. © 1997 by the National Council for Teachers of Mathematics. All rights reserved.
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Areas and Brownies
Solutions
Problem H1
Using the fact that each of the tangram pieces can be divided into some number of small triangles, we get the following areas: The small square, the parallelogram, and the dmedium triangle each have areas of 2 square units, and the large triangles each have areas of 4 square units. The area of the entire tangram is 16 square units.
Problem H2
The trapezoid is divided into two triangles, each with height h, the height of the trapezoid. One triangle has base b_{1} while the other has base b_{2}:
Since the area of a triangle is , the total area is , or , which is also the area of the trapezoid.
Problem H3
Using the Pythagorean theorem, a^{2} + b^{2} = c^{2}, we can calculate the lengths of the bases and the height. The bases have lengths of 2√2 (the hypotenuse of the triangle with legs 2 and 2) and 6√2 (the hypotenuse of the triangle with legs 6 and 6), respectively. Similarly, the height is 2√2 (it is the length perpendicular to the bases and the hypotenuse of the triangle with legs 2 and 2). The area is , which equals 16 square units.
We can also find the area by drawing a rectangle around the trapezoid and then subtracting the smaller areas, or by dividing the trapezoid into two triangles, or by any of several other methods. But any way you slice it, the area is 16 square units.
To learn more about the Pythagorean theorem, go to Learning Math: Geometry, Session 6.
Problem H4

 Both shapes have areas of 4.5 square units.

Number of Boundary Dots
Area
(in Square Units)3 0.5 4 1 5 1.5 6 2 7 2.5 b – 1 In each case, the area is half the number of boundary dots minus 1. If there are b boundary dots and one interior dot, the area is _{} 1.

Number of Boundary Dots
Area
(in Square Units)3 1.5 4 2 5 2.5 6 3 7 3.5 b In each case, the area is half the number of boundary dots. If there are b boundary dots and one interior dot, the area is .

Number of Boundary Dots
Area
(in Square Units)3 2.5 4 3 5 3.5 6 4 7 4.5 b + 1  In each case, the area is half the number of boundary dots plus 1.
If there are b boundary dots and two interior dots, the area is + 1.
Every time you add a boundary dot, the area goes up by half a unit, regardless of the orientation of the shape.  The formula is A = I + – 1, and it can be used on many of the figures in Part B of this session.