G.GMD.1 Wkst 2

G.GMD.1 STUDENT NOTES WS #2
1
THE RECTANGLE
The most basic two dimensional figure is the rectangle. Due to its perpendicular adjacent sides we are able to
determine the number of ‘square’ units within the shape by multiply the two dimensions of the rectangle
together. This results in the formula, Area = (length)(width) or Area = (base)(height). These four terms are
important to define and so we will do that now.
Length of a Rectangle – the longer dimension of the rectangle.
A = lw
Width of a Rectangle – the shorter dimension of the rectangle.
width
length
Base of a Rectangle – a base is a side of the rectangle.
A = bh
Height of a Rectangle – the height is the perpendicular adjacent side to
the base of the rectangle.
The truth is that because the sides of a rectangle are perpendicular,
these values are interchangeable.
height
base
A = bh
base
height
A BASE AND HEIGHT MUST ALWAYS BE PERPENDICULAR TO EACH OTHER.
1 cm
4 cm
2 cm
5 cm
6 cm
8 cm
Area = (6)(2) = 12 cm2
Area = (8)(4) = 32 cm2
Area = (5)(1) = 5 cm2
AREARECTANGLE = bh
THE PARALLELOGRAM
The parallelogram formula relies heavily on the area of a rectangle.
d
The general parallelogram has adjacent sides that are not
Area ≠ cd
perpendicular so we cannot simply do as we did for the rectangle by
multiplying the two adjacent sides. The two measurements are not
c
perpendicular to each other and so it would not determine the
number of ‘square’ units within the shape. We do see though a nice way to alter a parallelogram to get a
rectangle. If we cut off the ‘slanted’ piece and move it over to the opposite side we can form a rectangle. The
process of cutting up shapes up into smaller pieces and moving them around to form a new shape is called
DISSECTION. Dissection is common way to investigate area relationships because the areas of the pieces
preserve the total area of the original shape.
G.GMD.1 STUDENT NOTES WS #2
2
PARALLELOGRAM AREA – DISSECTION
The most common method for determine the area formula for a parallelogram is to dissect the shape as
follows.
height
height
base
base
height
base
We perform a translation of the triangle to its opposite side. This changes our parallelogram into a
rectangle with the same area. The new rectangle has the exact same base and height as the original
parallelogram and so the formula for a parallelogram is the same as a rectangle, Area = (base)(height) = bh.
AREAPARALLELOGRAM = bh
PARALLELOGRAM AREA – SHEARING
Another way to determine the area formula for a rectangle is to use a technique that is also very helpful
when working with area called SHEARING. Shearing is a transformation in which all points in one line
remain fixed while all other points move parallel to the fixed line by a proportional distance. So in the
case of a parallelogram if we fix one side and then translate the opposite side along a parallel path to the
fixed side we shear the shape. The advantage of this process is that the base and height of the
parallelogram remain the shape even though the shape is changing. If we do this process until the adjacent
sides of the parallelogram are perpendicular, then we have transformed our parallelogram into a rectangle
that has the same base and height. This of course means that the areas of these two figures are the same
and their formulas must also be the same, Area = (base)(height) = bh.
height
height
base
height
base
base
Given the following rectangles and parallelograms, calculate their areas.
(Lines that appear to be parallel or perpendicular are.)
10 cm
5 cm
2 cm
12 cm
5 cm
30°
2 cm
15 cm
7.5 cm
3 cm
12 cm
Area = ___________
Area = _____________
Area = _____________
Area = _____________
A = bh
A = 2(5) = 10 cm2
h = 6 cm
A = 15(6) = 90 cm2
32 + h2 = 52
h=4
A = bh
A = 2(10) + 5.5(2)
A = 31 cm2
A = bh
A = 15(4) = 60 cm2
G.GMD.1 STUDENT NOTES WS #2
3
Shearing a parallelogram is one way to preserve the area of the shape and yet allows us to transform it into a
rectangle so that area is easy to calculate.
Shear this parallelogram to determine its area.
Original Parallelogram
Three Stages of Shearing
5 cm2
2.5 cm
2 cm
*o*rt
G.GMD.7 WORKSHEET #2
1. Henry looks at the rectangle on the right and
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that the base is 5 cm and the
height is 3 cm. Jennifer looks at it and sbys that the base is 3 cm and the height is
5 cm. Who is correct? Explain.
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2. Demonstrate how using dissection the given parallelogram has the same area as a rectangle with the
same base and height.
base
3. ln the previous question, which transformation moved the dissected piece into its new location to form
the rectang le? Tvw^-s
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G.GMD.I WORKSHEET #2
5. Determine the area of the following figures. (Lines that appear to be perpendicular are perpendicular and
lines that appear to be parallel are.)
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G.GMD.I
WORKSHEET #2
3
7. Determine the area of the following rectangles and parallelograms. (Lines that appear to be perpendicular
are perpendicular and lines that appear to be parallel are.)
b)
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c)
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