6th Grade Math Textbook, Fundamentals

298 Chapter 10

Enrichment:

Area of Irregular Polygons

Objective To explore various methods for finding the area of irregular polygons

There are several ways to find
the area of an irregular polygon
if you can locate its vertices on
a coordinate grid. You will use
three of these methods to find
the area of hexagon ABCDEF
at the right.

Method 1
Subtract From a Rectangle

The steps below describe how
to find the area of an irregular
figure by forming a rectangle,
finding its area, and then
subtracting the area of the shapes
surrounding the irregular figure
from the area of the rectangle.

Vertices of

Hexagon ABCDEF

A(4, 3)

B (7, 5)

C (6, 1)

D(3, 1)

E (1, 4)

F(2, 7)

012345 6 7

x

B

D C

E

F

A

7 6 5 4 3 2 1

y

012345 6 7

x

B

D C

E

Q F

T

R

S

(^123)
(^45)
7 6
A
7 6 5 4 3 2 1
y
Enclose the
polygon in a
rectangle so that
as many polygon
vertices as
possible are on
the rectangle.
Make and label
right triangles
and rectangles as
needed to fill the
area outside the
polygon.
Use the grid to find
the area of the
large rectangle and
other figures
outside the polygon
and inside the
rectangle.
Subtract the sum
of the areas of the
figures outside
the polygon from
the area of the
large rectangle.
Here is how you would use this method
for finding the area of hexagon ABCDEF.
Draw a rectangle QRSTto enclose the
hexagon ABCDEF.
Form 1, 2, 3, 4, 5, 6, and 7 within
the rectangle QRSTthat are outside the
hexagon ABCDEF.

• Area of rectangle QRST:
  AQRST6 units • 6 units 36 units^2


• Areas of polygons within the rectangle QRST
  that are outside the hexagon ABCDEF:
  A 1 1.5 units^2 A 2 1 unit^2 A 3 8 units^2 A 4 1 unit^2
  A 5 3 units^2 A 6 2 units^2 A 7 3 units^2
  Total area of polygons: 19.5 units^2
  Area of hexagon ABCDEF:
  area of QRSTtotal area of polygons 36 unit^2 19.5 units^2
  16.5 units^2