Download Regular Polygons

Name: __________________________
What You’ll Learn



Properties of various polygons
How to calculate for the sum of interior angle in a regular polygon
How to calculate for the measure of an interior angle in a regular polygon
Why It’s Important
Polygons are used by:



Professional tilers creating designs for flooring or backsplashes.
Engineers looking to keep soldiers safe.
Brick layers for pathways.
Key Formulas
Sum of the interior angles:
NOTE: Here n = the number of sides
𝑆 = 180(𝑛 − 2)
Measure of an interior angles: 𝑀 =
Measure of central angle:
180 (𝑛−2)
𝑛
360
𝐶= 𝑛
The practice problems have been taken from a variety of sources including:
MathWorks 12
Math at Work 12
Apprenticeship and Workplace 12
Provincial exams
Grade 12 Essentials - Polygons
Getting Started: Notes
Measuring Angles
To measure an angle:
-
Put the vertical marker of the protractor at the
vertex (corner) of the angle you are measuring.
Make sure that the 0̊ line is along one of the legs of
the angle.
Follow out to the second leg and read the
measurement on the protractor. Depending on the
length of the line, it may be difficult to use the
outside measurements on the protractor.
Marking Sides and Angles
-
Capital letters are used to mark vertices of a polygon.
The line segment (side) of a polygon is denoted by the two vertices
(corners) it sits between.
Congruent (equal) sides are marked with dashes.
Congruent (equal) angles are marked with arcs.
Examples:
1. Congruency of Angles
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2. Congruency of Sides
Grade 12 Essentials - Polygons
Getting Started: Practice
NOTE:  is the symbol for angles. The letter that is listed in the middle is the angle
that is being measured.
Ex.  ABC = 60˚ means that angle B = 60˚. (We could also call this B)
1. Measure each of the following angles.
a)
A
B
ABC=
C
b)
ACB=
B
BCD=
A
D
C
2. Record the side lengths and angles for ∆ABC.
AB =
A=
AC=
B=
BC=
C=
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Grade 12 Essentials - Polygons
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Grade 12 Essentials - Polygons
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Grade 12 Essentials - Polygons
Triangles: Notes
Definitions
Triangle:
Vertex (pl. Vertices):
Classification
Triangles can be classified by their…

Angles
1. Right:
2. Acute:
3. Obtuse:

Side Lengths
1. Equilateral:
2. Isosceles:
3. Scalene:
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Grade 12 Essentials - Polygons
Triangles: Practice
1. Circle the types of triangles that have each property.
NOTE: There may be more than one right answer.
a) Some sides are equal.
Equilateral
Isosceles
Scalene
b) No interior angles are equal.
Equilateral
Isosceles
Scalene
c) The sum of the interior angles is 180˚.
Acute
Right
Equilateral
The Esplanade Riel Bridge
2. Circle the types of triangles that do not have each property.
a) All angles are less than 90˚.
Acute
Right
Obtuse
b) Some angles are equal.
Equilateral
Isosceles
Scalene
c) There are at least 2 equal sides.
Equilateral
Isosceles
Scalene
3. Use the angle measures to calculate the unknown angles in each triangle.
NOTE: The following are not to scale. Use calculations rather than a
protractor to solve.
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Grade 12 Essentials - Polygons
4. Use the following diagram to answer the questions below.
a) What is the measure of M?
b) Classify ∆MNP by angle measure and by side length.
5. Recall our activity at the start of class …
a) The sum of the interior angle plus the exterior angle is the same at
each vertex. What is this sum?
b) Why does it make sense that each vertex has the same sum?
c) Is this a property for all triangles? Explain.
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Grade 12 Essentials - Polygons
Quadrilaterals: Notes
Which of the following shapes are polygons? Circle each one.
Definitions
Polygon:
Quadrilateral:
Types of Quadrilaterals
Rectangle:
Square:
Parallelogram:
Rhombus:
Trapezoid:
Kite:
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Grade 12 Essentials - Polygons
Irregular Quadrilateral:
Concave Quadrilateral:
NOTE: Some polygons are also convex, which means that there are no interior
angles which are greater than 180˚.
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Grade 12 Essentials - Polygons
Activity – Properties of Quadrilaterals
We can take a polygon and draw diagonals, which are line segments joining
vertices that are NOT NEXT TO EACH OTHER.
Example) In pairs, complete the following instructions and answer each question
using the given square.
A
B
1. Draw the diagonal AC. Measure its length.
2. Draw the diagonal BD. Measure its length. What do
you notice?
3. Label the point where the diagonals intersect as E.
Measure the lengths of AE, BE, CE, and DE. What do
you notice?
D
C
4. Measure DEA, AEB, BEC, and CED. What do
you notice?
5. What is the sum of the angles where the diagonals
intersect?
What do we find?
-
The diagonals are
. That is, they are the equal.
The diagonals on a square are
. That is, they cross at a
90˚ angle.
The diagonals
each other. That is, they cut each other in half.
All regular polygons have certain properties when you draw diagonals. See the
following page for an overview of the properties.
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Grade 12 Essentials - Polygons
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Grade 12 Essentials - Polygons
Quadrilaterals: Practice
1. Determine the missing measurements using the properties of
quadrilaterals.
2. State two properties that would prove that a quadrilateral is a
parallelogram.
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Grade 12 Essentials - Polygons
3. List all the quadrilaterals that could fit each description.
a) Has at least one set of parallel sides
b) Has four equal side lengths
c) Has two equal diagonals
4. Sketch and name a quadrilateral that fits each description.
a) The diagonals are equal, but the sides are not all equal.
b) The diagonals are equal, and all the sides are equal.
c) The diagonals are not equal, and no two sides are equal.
5. Using your knowledge of the properties of quadrilaterals, find the
measures of the missing angles. What kind of quadrilateral is this?
6. Solve for the indicated length or angle, and identify the type of
quadrilateral.
a)
b)
a.
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Grade 12 Essentials - Polygons
Regular Polygons: Notes
Definitions
Regular Polygon:
Example 1) Are the following shapes regular
polygons? Explain.
a)
b)
Common Polygons
Name
Number of Sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Activity – Measure and Sum of Interior Angles
1. Draw a square. Then draw a diagonal between two non-adjacent corners
so the square is divided into triangles.
2. How many triangles were created?
3. What is the sum of the interior angles of a square?
4. What is the measure of each interior angles of a square?
5. For the regular pentagon below, repeat steps 2-5.
NOTE: You will need to draw more than one diagonal to divide each
shape into triangles.
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Grade 12 Essentials - Polygons
6. For the regular hexagon below, repeat steps 2-5. Again, you will need to
draw more than one diagonal to divide each shape into triangles.
7. Using your results from steps 1-7, complete the following table:
Figure
Number of
Sides
Number of
Triangles
Sum of Interior
Angles
Measure of Each
Individual Angle
Equilateral
Triangle
Square
Regular
Pentagon
Regular
Hexagon
8. Use your chart from Question 8 to answer the following questions.
a. How does the number of triangles you can make in a polygon
relate to the number of sides?
b. How many triangles can you make in a 12-sided polygon?
c. What is the sum of all the angles measures in a 12-sided polygon?
d. What is the measure of each angle in a 12-sided polygon?
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Grade 12 Essentials - Polygons
Properties
We can use formulas to find the measure of an interior
angle, as well as the sum of the interior angles of a
regular polygon.
Sum of the interior angles:
𝑆 = 180(𝑛 − 2) where n is the number of sides
Example 2) Find the sum of the interior angles of a hexagon.
Example 3) Working backwards: The sum of the interior angles of a polygon is
900˚. Determine the number of sides of the polygon.
Measure of an interior angle:
If we know that all of the angles in a hexagon sum to 720˚, how can we find one
angle?
𝑀=
180 (𝑛−2)
where n is the number of sides
𝑛
Example 4) Find the measure of an interior angle in a square.
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Grade 12 Essentials - Polygons
Measure of the central angle:
We can also determine the measure of the central angles in a regular polygon.
The central angle is the angle made at the center of a polygon by any two
adjacent vertices of the polygon.
All central angles would add up to 360º (a full circle), so the measure of the
central angle is 360 divided by the number of sides
360
𝐶 = 𝑛 where n is the number of sides.
Example 5) What is the measure of the central angle in a hexagon?
Example 6) A regular polygon has central angles of 45º.
a) State the number of sides for this polygon.
b) State the name of this polygon.
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Grade 12 Essentials - Polygons
Regular Polygons: Practice
1. Which are regular polygons? Check with a ruler and a protractor.
2. What is the measure of each interior angle in a regular octagon?
3. Given the following regular polygon:
a) Calculate the sum of the interior angles in the
polygon.
b) State the measure of each interior angle in the polygon.
4. The sum of the interior angles is 900º. Determine the number of sides of the
polygon.
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Grade 12 Essentials - Polygons
5. A regular hexagon has a side length of 10 metres.
a) State the measure of angle A, the central angle, in degrees.
b) State the measure of the given diagonal in metres.
6. Draw ALL diagonals in each regular polygon.
a) How many diagonals does each polygon have?
b) Decide whether each property is true or false using the above
polygons.

If the number of vertices is odd, the number of diagonals is odd.

If the number of vertices is even, the diagonals that connect opposite
vertices intersect at the centre.

The number of diagonals you can draw from one vertex of a regular
polygon is (𝑛 − 3), where 𝑛 is the number of vertices.
7. Determine the number of diagonals in a regular octagon.
NOTE: Use the formula 𝐷 =
20
𝑛(𝑛−3)
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Grade 12 Essentials - Polygons
Applications of Polygons
Airless Tire Promises Grace under Pressure for Soldiers
The Pentagon investigates the use of a new type of airless
tire designed to get troop-carrying Humvees through hot
spots without stopping
In Iraq and elsewhere, improvised explosive devices (IEDs)
pack a double-deadly whammy: They can kill when they
explode, and then they turn surviving soldiers into sitting
ducks when Humvee tires blow out. Conventional Humvee tires need a certain amount
of air pressure, but also may include so-called "run-flat" inserts that wrap around the
tire's rim to keep it from going completely flat when the tire's surface is ruptured. The U.S.
Army, however, is looking for an alternative that can keep its vehicles running faster and
farther than a run-flat donut after an attack.
To keep troops from being stranded and easily ambushed on the battlefield, the Army is
working with researchers to develop tires for their Humvees that can better withstand
roadside attacks. One such design comes from Resilient Technologies, LLC, based in
Wausau, Wisc., and the University of Wisconsin–Madison's Polymer Engineering Center.
With a four-year, $18-million grant from the Pentagon, Resilient is working to create a
"non-pneumatic tire" (NPT) technology, called that because it doesn't require air.
The NPT looks like a circle of honeycombs bordered by a thick black tread. "There's a lot
of space for shrapnel to pass through," says Ed Hall, Resilient Technologies's director of
business affairs. "Even if you remove 30 percent of the webs, the tire will still work."
And for those of you wondering why all tires aren't simply made out of solid rubber,
some construction vehicles use them on sites with debris that can easily shred a
pneumatic tire, but solid tires give an incredibly rough ride, generate a lot of heat, and
might be even worse if a piece came off during an explosion, because it could not
easily be repaired. The NPT's honeycomb structure is designed to support the load
placed on the tire, dissipate heat and offset some of these issues.
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Grade 12 Essentials - Polygons
Applications: Article Analysis
Based on the article “Airless Tire Promises Grace under Pressure for Soldiers”, fill
out the following chart.
Title and topic of article
Fact-Based Article Analysis
Summarize the main ideas in your own
words.
Draw a diagram to represent the main
idea of the article (i.e. an airless tire)
List, in point form, at least three facts in
the article.
Write one question you have from the
article.
This article is important because …
Polygons are often used in construction, commercial, industrial or
artistic applications. Come up with one other real world application
for polygons.
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Grade 12 Essentials - Polygons
Chapter Review
Definitions: You will not be asked to specifically define a term on the test.
However, this is vocabulary heavy unit. Make sure you know all of the different
terms used so that you understand the questions being asked. See the Polygon
Unit Vocabulary at the end of this booklet.
Short Answer/Problem Solving:
1. What type of triangle has two equal sides and all angles are less than 90°?
2. Is A congruent with
reasoning.
B? What type of triangles are they? Explain your
3. What type of triangle is
XYZ? Explain using
XYW and
YZW.
4. Calculate all unknown interior and exterior angles using the given angles.
5. Calculate all unknown interior and exterior angles using the given angles.
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Grade 12 Essentials - Polygons
6. The sum of the interior angles of a polygon is 540̊. Determine the number of sides
of the polygon.
7. Determine the unknown measurements using the properties of quadrilaterals.
a) Isosceles trapezoid
TW = 8
UV =
TV=12
UW=
TXW = 60°
UXV =
b) Square
OP = 8
OQ = 10
PQ =
PS =
PSO =
OSQ =
8. Sketch a rhombus and label ALL of the congruent parts.
9. Which of these quadrilaterals do not have equal angles at opposite vertices?
10.
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Grade 12 Essentials - Polygons
10. Use properties of quadrilaterals to show that a square is always a parallelogram,
but a parallelogram is not always a square.
11. Is this shape a regular polygon? Explain.
12. Is this shape a regular polygon? Explain.
13. What is the relationship between the number of triangles that can be formed
within a regular polygon and the sum of all angle measures? Explain.
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Grade 12 Essentials - Polygons
14. Determine the sum of the angles in each regular polygon. Then, state the
measure of each interior angle.
a) Pentagon
b) Octagon
c) 10-sided figure
d) 11-sided figure
15. Louis wants to put a hole in the centre of his patio table for a large sun umbrella.
The table is shaped like a regular decagon with ten equal sides. How can Louis
determine the location of the centre? Explain with an illustration.
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Grade 12 Essentials - Polygons
16. Given a regular hexagon with centre C.
a) Determine the measure of the central angle of the
hexagon.
b) Determine the length of side a. Justify your answer.
17. Determine (by illustration of calculation) the total number of diagonals in a
regular six-sided polygon.
NOTE: The following question will be asked on the test!
18. Polygons are often used in construction, commercial, industrial, or artistic
applications.
a) Demonstrate one use of the various properties of polygons in the real
world by performing the following two steps:


State a specific example where the various properties of polygons
are used.
Support your example with a written explanation of how various
properties of polygons are used.
b) Sketch a reasonable neat picture or diagram (not necessarily to scale)
that supports your example in Part A.
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Grade 12 Essentials - Polygons
Polygon Unit Vocabulary
Term
Definition
polygon
a closed shape made up of
straight lines
equilateral triangle
a triangle with three equal
sides
isosceles triangle
a triangle with exactly two
equal sides
scalene triangle
a triangle with no equal
sides
acute triangle
a triangle with each angle
less than 90o
obtuse triangle
a triangle with one angle
that is greater than 90o
right triangle
a triangle with one angle
that is equal to 90o
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Diagram
Grade 12 Essentials - Polygons
regular polygon
congruent
a closed shape with all
sides equal and all angles
equal
the same size and shape
(they are equal)
complementary angles
two angles whose sum is
90o
transversal
a line that intersects two or
more lines
opposite angles
non adjacent angles that
are formed by two
intersecting lines
supplementary angles
two angles who sum is 180 o
quadrilateral
convex polygon
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Grade 12 Essentials - Polygons
concave polygon
diagonal
bisect
perpendicular
Quadrilaterals:
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Triangles: