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In this unit you will learn what a polygon is and isn’t. You will investigate shapes that tessellate a plane and shapes that do not. You will also compute area and find dimensions of triangles and various types of quadrilaterals.
Polygons Tessellations
Area of Parallelograms
Area of Other Polygons Summary of Area Formulas
polygon - A polygon is a closed figure that is made up of line segments that lie in the same plane. Each side of a polygon intersects with two other sides at its endpoints.
Here are some examples of polygons:
Here are some examples of figures that are NOT classified as polygons. The reason the figure is not a polygon is shown below it.
Example 1 : Is the shape a polygon? Explain why or why not.
The shape is not a polygon since the path is open.
Polygons
Not Polygons
The path is open.
Two sides of the figure intersect at a point other than the endpoints.
One side of the figure is curved.
In general, a polygon with n sides is called an n-gon. Several common polygons have been given names based on the number of sides.
regular polygon – A regular polygon is a convex polygon with all sides and angles congruent.
S = 180( n – 2) S represents the sum of all interior angles. n represents the number of sides in a polygon.
Example 3 : Draw a pentagon and draw all possible non-overlapping diagonals from one vertex, and then answer the following questions.
(a) How many triangles are formed?
Three triangles are formed.
Number of Sides Polygon 3 4 5 6 7 8 9
n
triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon n -gon
In a convex polygon with n sides, the sum of its interior angles equals 180( n - 2) degrees. Theorem 25-A
(b) Apply the Triangle Sum Theorem to determine the total number of degrees for the five angles in a pentagon.
Three triangles are formed by the non-overlapping diagonals. The sum of the angles in three triangles formed is equal to the sum of all five angles in the pentagon. Thus, the following statements can be made:
180 180 180 Results of dividing the pentagon into triangles. 180(3) Meaning of Multiplication 540 Simplify
The sum of the five angles in a pentagon is 540 degrees.
(c) Now apply Theorem 25-A to solve the same problem. 180( 2) Theorem 25-A 180(5 2) Substitution (A pentagon has 5 sides.) 180(3) Simplify 540 Simplify
S n S S S
By theorem 25-A, we have determined that the sum of the five angles in a pentagon is 540 degrees.
The formula stated in Theorem 25-A simplifies the method to determine the sum of the angles in any convex polygon.
Example 4 : What is the sum of the angles in a dodecagon? Since there are twelve sides in this polygon, we will use Theorem 25-A to shorten our work.
180( 2) Theorem 25-A 180(12 2) Substitution (A dodecagon has 12 sides.) 180(10) Simplify 1800 Simplify
S n S S S
The sum of the twelve angles in a dodecagon is 1800 degrees.
Angles Y and U 7 1 Expression for and. 7(17) 1 Substitution 120 Simplify
x + ∠ Y ∠ U
Angles Y and U measure 120 degrees.
Angles X and V 9 11 Expression for and. 9(17) 11 Substitution 142 Simplify
x − ∠ X ∠ V −
Angles X and V measure 142 degrees.
Check: 2(98) + 2(120) + 2(142) = 720 The six angles total 720 degrees.
Example 6 : What is the size of an interior angle of a regular dodecagon?
180( 2) Theorem 25-A 180(12 2) Substitution (A dodecagon has 12 sides.) 180(10) Simplify 1800 Simplify
(^1800) Divide by 12 for the twelve congruent angles. 12
Each angle equals 150 degrees.
S n S S S
The size of an interior angle in a regular dodecagon is 150 degrees.
Example 7 : For the regular octagon shown below, answer each question.
(a) What is the size of one interior angle ( x )?
180( 2) Theorem 25-A 180(8 2) Substitution (An octagon has 8 sides.) 180(6) Simplify 1080 Simplify
(^1080) Divide by 8 for the eight congruent interior angles. 8
One interior angle measures 135 degrees;
S n S S S
x
= 135 degrees.
(b) What is the size of one exterior angle ( x )?
180 and are linear angles. 135 180 Substitution ( 135 from part a) 45 Subtraction
One exterior angles measures 45 degrees; 45 degrees.
x y x y y x y
y
(c) What is the sum of all the exterior angles? 45(8) There are 8 exterior angles in an octagon. 360 Simplify
The sum of the exterior angles of a regular octagon is 360 degrees.
x ° y °
tessellation – A tessellation is a complete pattern of repeating shapes or figures that cover a plane leaving no spaces or gaps.
A tessellation may be created using slides, flips, and turns. Interesting tessellations may be formed beginning with a square or equilateral triangle.
Example 1 : Create a figure from a square that will tessellate a plane.
Start by drawing a square. Cut a triangle from its side and slide the triangle to the opposite side. (Use tape to attach the triangle.)
Sketch a face on the left side, cut it out and slide it to the opposite side.
Add additional features to make the figure more interesting.
A design is created that tessellates a plane.
Example 2 : Create a figure from an equilateral triangle that will tessellate a plane.
Draw an equilateral triangle and cut out a shape from one vertex and then rotate it around the vertex to another side.
Tessellate the plane.
You can tessellate a plane with regular polygons when they form a 360 angle at their touching vertices.
Example 5 : Give an example of a regular polygon that may be used to create a regular tessellation.
A regular hexagon tessellates the plane. At any one vertex of the tessellation, three angles of a hexagon, each measuring 120 degrees, meet. 120(3) = 360
Therefore, hexagons tessellate a plane and are an example of a regular tessellation.
semi-regular tessellation – A semi-regular tessellation is a tessellation that contains two or more regular polygons that tessellate the plane.
Example 6 : What combination of regular shapes tessellates a plane?
A regular octagon by itself will not tessellate a plane; however, combine it with a square that is turned to give the appearance of a diamond shape and the two together tessellate the plane. Let’s examine why this is true. At any one vertex of the figure, two angles from the two octagons and an angle from the square meet. Each of the angles in the regular octagon measure 135 degrees. The angle from the square measures 90 degrees. 135 + 135 + 90 = 360
The combination of regular octagons and squares will tessellate the plane when they are arranged such that three angles meeting at one vertex total 360 degrees. The tessellation is an example of a semi-regular tessellation.
uniform tessellation – A uniform tessellation is a tessellation that has the same combination of shapes and angles at each vertex.
The figure below is an example of a uniform tessellation.
The area of a parallelogram can be rearranged into the shape of a rectangle if the parallelogram is cut along an altitude (perpendicular height) from the top to its base.
Given: Parallelogram with a base of 10 units and a height of 8 units.
Cut the parallelogram along the height.
Rearrange the area into the shape of a rectangle by sliding the piece that is right of the altitude to the left, aligning it with the left side of the parallelogram.
The amount of coverage, the height, and the base all remain the same even after the rearrangement of the pieces of the parallelogram. Thus, the formula for finding the area of a parallelogram is the same as finding the area of a rectangle. Just remember to use the height of the parallelogram as the height, not the length of the slanted sides.
The area of a parallelogram is the product of its base and height.
A = bh
Height = 8 units
Base = 10 units
Height = 8 units
Base = 10 units
Note: The height of a parallelogram is shorter than the length of the slanted side. Be sure to measure the height of a parallelogram, not its slanted side, when determining is area.
Example 1 : Find the area of a parallelogram with a base of 10 units and a height of 8 units.
Formula for area of parallelogram 10(8) Substitution 80 square units Multiply
A b h A A
Height = 8 units
Base = 10 units
Example 3 : Find the area of parallelogram FGHJ.
We know that the formula for finding the area of a parallelogram is A = bh.
We are given the base (25 cm), but must calculate the height.
Triangle FKJ is a 45-45-90 degree right triangle. Recall that the legs of this type of triangle are equal in length and that the hypotuse equals the length of one leg times 2.
(Hypotenuse) 8 Given
Let length of.
8 2 The hypotenuse length of one leg times 2.
(^8) Division Property 2
(^8 2) Multiply by 2. 2 2 2
4 2 Simplify
x KJ
x
x
x
x
x
Thus, the height of the parallelogram is 4 2.
8 cm
25 cm
Now we have all that we need to find the area of the parallelogram.
Formula for area of a parallelogram. 25 4 2 Substitution 100 2 Simplify by multiplying the whole numbers outside the radical.
The area of the parallelogram is 100 2 square centimeters. *We can expr ess
A b h A A
the answer as a radical or evaluate it as a rounded decimal (141.4 sq cm).
Example 4 : Find the amount of tiling that would be needed to cover the two bath areas for the given floor plan.
Study the floor plan and notice that the width of the bedroom and the two baths is 25 feet. Also notice that the width of the bedroom is 16 feet.
Therefore, the width of the two baths:
25 − 16 =9 ft
Examining the floor plan closely, notice that the length of the two baths is given as 12 feet.
Garage
20 ft
24 ft
70 ft
Closet Closet
Bedroom
Bedroom (^) Bedroom
Living Room
Kitchen
Bath
Bath
Deck
13 ft^ 25 ft
9 ft
12 ft
25 ft
16 ft
