1. Polygons and Area § 10.1 Naming Polygons
§ Diagonals and Angle Measure
§ Areas of Polygons
§ Areas of Triangles and Trapezoids
§ Areas of Regular Polygons
§ Symmetry
§ Tessellations

2. Vocabulary What You'll Learn
Naming Polygons
What You'll Learn
You will learn to name polygons according to the number of
_____ and ______.
sides
angles
Vocabulary
1) regular polygon
2) convex
3) concave

3. A polygon is named by the number of its _____ or ______. sides angles
Naming Polygons
A polygon is a _____________ in a plane formed by segments, called sides.
closed figure
A polygon is named by the number of its _____ or ______.
sides
angles
A triangle is a polygon with three sides. The prefix ___ means three.
tri

4. Prefixes are also used to name other polygons. Prefix Number of Sides
Naming Polygons
Prefixes are also used to name other polygons.
Prefix
Number of
Sides
Name of
Polygon
tri- 3
triangle
quadri- 4
quadrilateral
penta- 5
pentagon
hexa- 6
hexagon
hepta- 7
heptagon
octa- 8
octagon
nona- 9
nonagon
deca- 10
decagon

5. Terms Naming Polygons A vertex is the point of intersection of
two sides.
Consecutive vertices are
the two endpoints of any
side. U T S Q R P
A segment whose
endpoints are
nonconsecutive
vertices is a
diagonal.
Sides that share a vertex
are called consecutive
sides.

6. An equilateral polygon has all _____ congruent. sides
Naming Polygons
An equilateral polygon has all _____ congruent.
sides
An equiangular polygon has all ______ congruent.
angles
A regular polygon is both ___________ and ___________.
equilateral
equiangular
equilateral
but not
equiangular
equiangular
but not
equilateral
regular,
both equilateral
and equiangular
Investigation: As the number of sides of a series of regular polygons increases, what do you
notice about the shape of the polygons?

7. A polygon can also be classified as convex or concave.
Naming Polygons
A polygon can also be classified as convex or concave.
If all of the diagonals
lie in the interior of
the figure, then the
polygon is ______.
If any part of a diagonal lies
outside of the figure, then the
polygon is _______.
concave
convex

8. Diagonals and Angle Measure
What You'll Learn
You will learn to find measures of interior and exterior angles
of polygons.
Vocabulary
Nothing New!

9. Diagonals and Angle Measure Number of Diagonals from One Vertex
Make a table like the one below.
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360

10. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex pentagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540

11. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex hexagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720

12. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex heptagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720
heptagon 7 4 5
5(180) = 900

13. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2
2(180) = 360
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720
heptagon 7 4 5
5(180) = 900
n-gon n
n - 3
n - 2
(n – 2)180
Theorem 10-1
If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

14. Diagonals and Angle Measure
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of any
convex polygon to form exterior angles.
48°
57°
74°
The figure suggests a method for finding the
sum of the measures of the exterior angles of a convex polygon.
72°
55°
54°
When you extend n sides of a polygon, n linear pairs of angles are formed.
The sum of the angle measures in each linear pair is 180.
sum of measure of
exterior angles
sum of measures of
linear pairs
sum of measures of
interior angles = – =
n•180 –
180(n – 2) =
180n –
180n + 360
sum of measure of
exterior angles =
360

15. Diagonals and Angle Measure
Theorem 10-2
In any convex polygon, the sum of the measures of the
exterior angles, (one at each vertex), is 360.
Java Applet

16. Vocabulary What You'll Learn
Areas of Polygons
What You'll Learn
You will learn to calculate and estimate the areas of polygons.
Vocabulary
1) polygonal region
2) composite figure
3) irregular figure

17. Any polygon and its interior are called a ______________.
Areas of Polygons
Any polygon and its interior are called a ______________.
polygonal region
In lesson 1-6, you found the areas of rectangles.
Postulate 10-1
Area Postulate
For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the polygon.
Area can be used to describe, compare, and contrast polygons. The two polygons below are congruent. How do the areas of these polygons compare?
They are the same.
Postulate 10-2
Congruent polygons have equal areas.

18. The figures above are examples of ________________. composite figures
Areas of Polygons
The figures above are examples of ________________.
composite figures
They are each made from a rectangle and a triangle that have been placed
together. You can use what you know about the pieces to gain information
about the figure made from them.
You can find the area of any polygon by dividing the original region into
smaller and simpler polygon regions, like _______, __________,
and ________.
squares
rectangles
triangles
The area of the original polygonal region can then be found by __________
_________________________.
adding the
areas of the smaller polygons

19. Area Addition Postulate
Areas of Polygons
Postulate 10-3
Area Addition Postulate
The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon. 1 2 3
AreaTotal =
A1 +
A2 + A3

20. Find the area of the polygon in square units.
Areas of Polygons
Find the area of the polygon in square units.
Area of Square
3u X 3u = 9u2
Area of Rectangle
1u X 2u = 2u2
Area of polygon =
= 7u2
3 units
Area of
Rectangle
Area of Square

21. Areas of Triangles and Trapeziods
What You'll Learn
You will learn to find the areas of triangles and trapezoids.
Vocabulary
Nothing new!

22. Areas of Triangles and Trapezoids
Look at the rectangle below. Its area is bh square units.
The diagonal divides the rectangle into two _________________.
congruent triangles
The area of each triangle is half the area of the rectangle, or
This result is true of all triangles and is formally stated in Theorem 10-3. b h

23. Areas of Triangles and Trapezoids
Height
Consider the area of this rectangle
A(rectangle) = bh
Base

24. Areas of Triangles and Trapezoids
Theorem
10-3
Area of a
Triangle
If a triangle has an area of A square units,
a base of b units,
and a corresponding altitude of h units, then h b

25. Areas of Triangles and Trapezoids
Find the area of each triangle:
6 yd
18 mi
23 mi
A = 13 yd2
A = 207 mi2

26. Because the opposite sides of a parallelogram have the same length, the area of a parallelogram is closely related to the area of a ________.
Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms.
rectangle
height
base
The area of a parallelogram is found by multiplying the ____ and the ______.
base
height
Base – the bottom of a geometric figure.
Height – measured from top to bottom, perpendicular to the base.

27. Areas of Triangles and Trapezoids
Starting with a single trapezoid. The height is labeled h, and the bases are labeled b1 and b2
Construct a congruent trapezoid and arrange it so that a pair of congruent legs
are adjacent. b1 b2 b1 h b2
The new, composite figure is a parallelogram.
It’s base is (b1 + b2) and it’s height is the same as the original trapezoid.
The area of the parallelogram is calculated by multiplying the base X height.
A(parallelogram) = h(b1 + b2)
The area of the trapezoid is one-half of the parallelogram’s area.

28. Areas of Triangles and Trapezoids
Theorem
10-4
Area of a
Trapezoid
If a trapezoid has an area of A square units,
bases of b1 and b2 units,
and an altitude of h units, then b1 b2 h

29. Areas of Triangles and Trapezoids
Find the area of the trapezoid:
18 in
20 in
38 in
A = 522 in2

30. Areas of Regular Polygons
What You'll Learn
You will learn to find the areas of regular polygons.
Vocabulary
1) center
2) apothem

31. Areas of Regular Polygons
Every regular polygon has a ______,
a point in the interior that is equidistant
from all the vertices.
center
A segment drawn from the center that is perpendicular to a side of the regular
polygon is called an ________.
apothem
In any regular polygon, all apothems are _________.
congruent

32. Areas of Regular Polygons
Now, create a triangle by drawing segments from the center to each vertex on
either side of the apothem.
Now multiply this times the number of triangles that make up the regular polygon.
The area of a triangle is calculated with the following formula:
The figure below shows a center and all vertices of a regular pentagon.
An apothem is drawn from the center, and is _____________ to a side.
perpendicular
There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 72
72°
72°
72°
72° s a
72°
What measure does 5s represent?
perimeter
Rewrite the formula for the area of a pentagon using P for perimeter.

33. Areas of Regular Polygons
Theorem 10-5
Area of a
Regular Polygon
If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then P

34. Areas of Regular Polygons
Find the area of the shaded region in the regular polygon.
5.5 ft
8 ft
Area of polygon
Area of triangle
To find the area of the shaded region, subtract the area of the _______
from the area of the ________:
triangle
pentagon
The area of the shaded region:
110 ft2 – 22 ft2 =
88 ft2

35. Areas of Regular Polygons
Find the area of the shaded region in the regular polygon.
Area of polygon
Area of triangle
6.9 m
8 m
To find the area of the shaded region, subtract the area of the _______
from the area of the ________:
triangle
hexagon
The area of the shaded region:
165.6 m2 – 55.2 m2 =
110.4 m2

36. Areas of Regular Polygons
Geometer's Sketchpad

37. Section Title
What You'll Learn
You will learn to
Vocabulary 1)

38. Section Title

39. Section Title
What You'll Learn
You will learn to
Vocabulary 1)

40. Section Title