12
Geometry
Copyright © Cengage Learning. All rights reserved.
12.2
Quadrilaterals
Copyright © Cengage Learning. All rights reserved.
Quadrilaterals
A parallelogram is a quadrilateral with opposite sides
parallel.
In Figure 12.19, sides
and
are parallel, and sides
and
are parallel. Polygon ABCD is therefore a
parallelogram.
Parallelogram
Figure 12.19
3
Quadrilaterals
Figure 12.20(a) shows the same parallelogram with a
perpendicular line segment drawn from point D to side
This line segment is an altitude.
Figure 12.20(b) shows the result of removing the triangle at
the left side of the parallelogram and placing it at the right
side.
You now have a rectangle with sides of lengths b and h.
(b)
(a)
Figure 12.20
4
Quadrilaterals
Note that the area of this rectangle, bh square units, is the
same as the area of the parallelogram.
So the area of a parallelogram is given by the formula
A = bh, where b is the length of the base and h is the length
of the altitude drawn to that base.
The perimeter is 2a + 2b or 2(a + b).
5
Quadrilaterals
A rectangle is a parallelogram with four right angles. The
area of the rectangle with sides of lengths b and h is given
by the formula A = bh. (See Figure 12.21.)
Rectangle
Figure 12.21
6
Quadrilaterals
Another way to find the area of a rectangle is to count the
number of square units in it. In Figure 12.22, there are 15
squares in the rectangle, so the area is 15 square units.
Figure 12.22
7
Quadrilaterals
The formula for the area of each of the following
quadrilaterals follows from the formula for the area of a
rectangle.
A square (Figure 12.23) is a rectangle with the lengths of
all four sides equal. Its area is given by the formula
A = b  b = b2. The perimeter is b + b + b + b, or 4b. Note
that the length of the altitude is also b.
Square
Figure 12.23
8
Quadrilaterals
A rhombus (Figure 12.24) is a parallelogram with the
lengths of all four sides equal. Its area is given by the
formula A = bh. The perimeter is b + b + b + b, or 4b.
A trapezoid (Figure 12.25) is a quadrilateral with only two
sides parallel. Its area is given by the formula
The perimeter is a + b + c + d.
Rhombus
Trapezoid
Figure 12.24
Figure 12.25
9
Quadrilaterals
Summary of Formulas for Area and Perimeter of
Quadrilaterals
Quadrilateral
Area
Perimeter
Rectangle
A = bh
P = 2(b + h)
Square
A = b2
P = 4b
Parallelogram
A = bh
P = 2(a + b)
Rhombus
A = bh
P = 4b
Trapezoid
P=a+b+c+d
10
Example 1
Find the area and the perimeter of the parallelogram shown
in Figure 12.26.
The formula for the area of a
parallelogram is
Figure 12.26
A = bh
So
A = (27.2 m)(15.5 m)
= 422 m2
11
Example 1
cont’d
The formula for the perimeter of a parallelogram is
P = 2(a + b)
So
P = 2(19.8 m + 27.2 m)
= 2(47.0 m)
= 94.0 m
12