AREAs of Polygon

1
AREAs of Polygon
2
CONCEPT OF AREA
rectangle
Rectangular region
The first figure above is a rectangle and the
second is a rectangular region.  
3
CONCEPT OF AREA
Rectangular region
rectangle
A rectangular region is a union of a rectangle
and its interior. When you are asked to find the
area of a rectangle, you are actually asked to
determine the area of a rectangular region.  
4
AREAS of a polygonal region

• - is the number of square units contained in the
  region.
• A square unit is a square with a side 1 unit in
  length.

5
AREAS of a polygonal region

• A unit for measuring area can be any shape we
  choose. But for convenience and by tradition, we
  use a square whose sides are a unit for measuring
  length, and refer to its area as one square
  unit

6
1 SQUARE unit
1 unit
1 unit
The standard units of area are square units, such
as square centimeters, square decimeters and
square meters.
7
Example 1
In the rectangle above, each small square is one
unit in length. Find the area of the
rectangle.  
The area can be determined by counting the number
of small squares. Since there are 24 small
squares, therefore, the area is 24 square units.

8
AREAs of Polygon
9
Area of a RECTANGLE
l
A lw where l is the length and w is the width.
w
w
l
10
Example 1
8 units
3 units
The area can be determined by counting the number
of small squares. Since there are 24 small
squares, therefore, the area is 24 square units.

Solution A lw 8(3) A 24 square units

11
Area of a SQUARE
A (s)(s) or A s² Where s is the length of
a side.
s
s
12
EXAMPLE 2
Find the area of the given figure.
Solution A s² (5 cm)² A 25 cm²
5 cm
5 cm
13
Area of a PARALLELOGRAM
h(height)
h w
b (base)
b l
In the figure, h is the height and b is the base.
Thus, Area of a //gram Area of a rectangle
lw
By substitution,
A bh
14
Example 3
Find the area of the given figure.
h 4 cm
b 6 cm
Solution Area of a //gram bh
(6 cm)(4 cm) A
24 cm²
15
Area of a triangle
h(height)
b (base)
In the figure, h is the height and b is the base.
Thus, Area of a triangle ½ Area of a
parallelogram A ½ bh
16
Find the Area of a triangle
Example 4
h 8 cm
b 10 cm
Solution A ½ bh
½ (10 cm) (8 cm)
½ (80 cm²) A 40 cm²
17
AREA OF A TRAPEZOID
What is a trapezoid? - A quadrilateral with one
pair of parallel sides.
The parallel sides are called the BASES.
The nonparallel sides are called the LEGS.
18
TRAPEZOID
b1
In the figure, b1 b2 are the bases and h is the
height . Note Height (h) is a segment drawn
from a vertex of any polygon ? to the opposite
side.
h
b2
19
HOW DO WE FIND THEAREA OF A TRAPEZOID?
b1
The formula to find the area of a trapezoid was
derived from the area of a triangle.
h
b2
20
Look at this..
Do you know why the height of the triangles are
the same?
If you draw a diagonal, what are the new figures
formed?
b1
I
h
II
b2
Triangle 1 and Triangle 2
Area of Triangle 1 ½b?h
Area of Triangle 2 ½b?h
21
AREA OF A TRAPEZOID
Area of a trapezoid is equal to the sum of the
areas of two triangles.
b1
I
h
II
b2
A Triangle 1 Triangle 2
A ½b?h ½b?h
A ½h(b? b?)
22
TRAPEZOID
b1
In the figure, b1 b2 are the bases and h is the
height. Thus, A ½h( b1 b2 )
h
b2
23
EXAMPLE 5
b1
In the figure, h 8 cm, b1 b2 are 4 cm 10 cm
respectively. Find the area of a trapezoid.
h
b2
24
Solution
4 cm
A ½h(b? b?)
½(8 cm)(4 cm 10 cm)
8 cm
½(8 cm)(14 cm)
(4cm)(14 cm)
10 cm
A 56 cm²
25
How much do you know
26
Find the area of the following 1. A square with
side of 25cm. 2. A parallelogram with base 17 m
and height 14 m. 3. A triangle with base equal to
12 cm and altitude equal to 10 cm. 4. A rectangle
17m by 11m. 5. A trapezoid with height 6 cm and
the length of the bases are 7 cm and 9 cm.
27
Find the area of the following figures.

• 1. A s²
• (25 cm) ²
• A 625 cm²

• 2. A bh
• (17 m)(14 m)
• A 238 m²

• 3. A ½bh
• ½ (12 cm)(10 cm)
• (6 cm )(10 cm)
• A 60 cm²

28
Find the area of the following figures.

• 4. A lw
• (17 m) (11 m)
• A 187 m²

• 5. A ½h(b1 b2)
• ½ (6cm)(7 cm 9 cm)
• (3 cm )(16 cm)
• A 48 cm²

29
Lets Summarize
30

• MAJOR CONCEPTS
• The area of a region is the number of square
  units contained in the region.
• 2. A square unit is a square with a side
  one (1) unit in length.
• 3. The area (A) of a rectangle is the product
  of its length (l) and its width (w). Thus,
• A lw

31
MAJOR CONCEPTS 4. The area (A) of a square is the
square of the length of a side (s). A s2 5.
The area (A) of a parallelogram is equal to the
product of the base (b) and the height (h).
Thus, A bh
32
MAJOR CONCEPTS 6. The diagonal separates the
parallelogram into two congruent triangles. 7.
The area (A) of a triangle equals half the
product of the base (b) and the height (h). Thus,
A ½bh. Sometimes altitude is used instead of
height.
33

• MAJOR CONCEPTS
• The area (A) of a trapezoid is one half the
  product of the length of its altitude and the
  sum of the lengths of two bases. Thus,
• A ½h (b1 b2).