Geometry: An Interactive Journey to Mastery

Lesson 21: Understanding Area

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x You can memorize that the area of a parallelogram is “base times height” or that the area of a rhombus

is “half the product of its diagonals,” and so on, if you wish. Be selective about which formulas you

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EHGHFRPSRVHGLQWRWULDQJOHV௘ͽ

 6KRZWKDWWKHDUHDRIDUKRPEXVHTXDOVKDOIWKHSURGXFWRI

the lengths of its diagonals.

 Five students were asked to write a formula for the

area A between two squares with dimensions as shown

in )LJXUH.

Albert thought of this shaded region as the union of four

3 × 3 squares and four 3 × xUHFWDQJOHVͼ௘6HH)LJXUH௘ͽ

Thus, he was compelled to write A = 12x + 36.

D௘ͽ %LOEHUWZURWHA ͼ௘x௘ͽ^2 íx^2. How was he viewing the

¿JXUHWREHOHGWRWKLVIRUPXOD"

E௘ͽ &XWKEHUWZURWHA îͼ௘x௘ͽ:KDWZDVKHVHHLQJWKDW

led him to write this formula?

F௘ͽ 'LOEHUWZURWHAx ˜  ˜ 43 6 49. What did he

visualize to see this formula as the natural answer to

the problem?

G௘ͽ (JEHUWZURWHA îͼ௘x௘ͽîx. What did Egbert see to lead him to this expression?

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Problems

33

3

3

3

3

xx

x

xx

x

x

33 x

Figure 21.6

33

3

3

3

3

xx

x

33 x

Figure 21.7