POLYGONS and AREA Classifying Polygons Angles in Polygons

POLYGONS and AREA Classifying Polygons Angles in Polygons
Area of Squares and Rectangles Area of Triangles Area of Parallelograms Area of Trapezoids Circumference and Area of Circles OPENERS Assignments Reviews

POLYGONS and AREA Classifying Polygons MENU POLYGON BASICS

POLYGONS and AREA Classifying Polygons
MENU When people use the word “SHAPE” they are usually referring to a POLYGON. So, what is a POLYGON? Basically, it is a CLOSED shape with “straight” sides

means the shape is complete
POLYGONS and AREA Classifying Polygons MENU CLOSED means the shape is complete closed NOT closed

POLYGONS and AREA Straight Sides No curves “Straight” Curved
Classifying Polygons MENU Straight Sides No curves “Straight” Curved

What do all these shapes have in common?
POLYGONS and AREA Classifying Polygons MENU What do all these shapes have in common? They are all simple polygons. To be a polygon you need 2 things: CLOSED STRAIGHT

MENU Why do we need to know about polygons? Polygons show up all over in nature, science, engineering…

POLYGONS and AREA Classifying Polygons If you play video games…
MENU If you play video games… You may have seen the word POLYGONS, and know it has something to do with graphics. This is because C.A.D. programs use polygons to render objects C. computer A. aided D. drafting Make a 3-D model

POLYGONS and AREA Classifying Polygons MENU

MENU POLYGONS are the basis of most computer imaging. The more POLYGONS and image has, the higher the quality

POLYGONS and AREA Classifying Polygons MENU Before we do anything with polygons, you must understand the difference between CONVEX and CONCAVE. Convex Concave

POLYGONS and AREA Classifying Polygons MENU It is easier to explain CONCAVE than it is to explain CONVEX A polygon is CONCAVE if: There are 2 points somewhere inside the shape So that if you connect those 2 points with a line The line goes outside the shape.

It may help to think of CONCAVE as having a “cave”
POLYGONS and AREA Classifying Polygons It may help to think of CONCAVE as having a “cave” or indentation MENU These are all CONCAVE

POLYGONS and AREA Classifying Polygons MENU If you cannot find 2 points that make a line that goes outside . . . Then the shape is CONVEX

MENU Another way to determine if a polygon is CONVEX or CONCAVE is to extend all the sides… And if any of the lines go back into the shape, then it is CONCAVE

CONVEX or CONCAVE? POLYGONS and AREA Classifying Polygons CONCAVE
MENU CONVEX or CONCAVE? CONCAVE CONCAVE CONCAVE CONVEX CONVEX CONCAVE CONVEX CONVEX CONCAVE

MENU These two aren’t even POLYGONS, why? They have curves.

POLYGONS and AREA Classifying Polygons MENU CONCAVE IS BAD!! In regular geometry, CONCAVE shapes are like your strange cousin. We just don’t talk about them. We will spend almost all of our time on CONVEX shapes

What do we call a polygon with three sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a polygon with three sides? A triangle

What do we call a shape with four sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a shape with four sides? A quadrilateral

What do we call a shape with five sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a shape with five sides? A pentagon

What do we call a shape with six sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a shape with six sides? A hexagon

What do we call a shape with seven sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a shape with seven sides? A heptagon (sometimes also called a septagon)

What do we call a shape with eight sides?
POLYGONS and AREA Classifying Polygons MENU What do we call a shape with eight sides? An octagon

A nine sided figure is called …
POLYGONS and AREA Classifying Polygons MENU A nine sided figure is called … A nonagon A ten sided figure is called … A decagon

POLYGONS and AREA Classifying Polygons MENU While the shapes with more than 10 sides have names, it is acceptable to call them “n-gons” What does THAT mean? It means you can call an eleven sided shape an “11-gon” and a twenty-three sided polygon a “23-gon”

All the sides are congruent,
POLYGONS and AREA Classifying Polygons MENU In a REGULAR polygon… EQUILATERAL All the sides are congruent, EQUIANGULAR All the angles are congruent 3 3 1080 1080 1080 3 3 1080 1080 3

POLYGONS and AREA Classifying Polygons MENU No matter how many sides the polygon has, they all have the same parts. VERTEX (vertices) SIDE

Naming any polygon has 1 simple rule:
POLYGONS and AREA Classifying Polygons MENU Naming any polygon has 1 simple rule: Pick any 1 vertex to start, Then go around the shape, clockwise OR counterclockwise heptagon BCDEFGA B heptagon FEDCBAG A C To keep it simple we will try to go as close to alphabetical as we can D G heptagon ABCDEFG F E

POLYGONS and AREA PERIMETER In other words: Classifying Polygons
MENU PERIMETER is the distance around the outside of a 2-D object In other words: If you walked around a polygon, how far would you walk?

MENU What is the perimeter of this rectangle? 12 4 4 12 Perimeter: = 32 OR 2(12 + 4)

MENU Find the perimeter for each of the following polygons: 7 1. 2. 3. 4 4 4 4 3 4 22 4 3 4 40 4 8 4 4 7 4 4 2 3 2 6. 4. 5. 11 1 1 11 1 1 5 3 3 55 20 7 11 11 2 1 1 1 1 22 2 3 11 4

MENU Find the perimeter for the following REGULAR polygons: 12-gon 7. 8. 24 40 2 8 9. 20 60

MENU 10. Find the perimeter of a regular nonagon with side lengths of 13? 117 11. Find the perimeter of a regular 27-gon with side lengths of 6? 162

Interior and Exterior angles in Polygons
POLYGONS and AREA Angles in Polygons MENU Interior and Exterior angles in Polygons

POLYGONS and AREA Angles in Polygons
MENU The INTERIOR ANGLES of a polygon are the angles inside the figure

POLYGONS and AREA POLYGON ANGLES Angles in Polygons
MENU POLYGON ANGLES We know the angles of a triangle add to 180. 180 In the other shapes, we draw in triangles to find the angle sum. 180 180 180 =540 =360 180 180 180 180 180 =720 180 =1080 180 180 180 180 180 180

INTERIOR ANGLES POLYGONS and AREA Angles in Polygons
MENU INTERIOR ANGLES THEOREM: The Sum of the INTERIOR angles of a convex polygon is (n-2) x 180. (n is the number of sides) So in a pentagon (5 sides), n=5 The sum of the interior angles: (5-2) x 180 3 x 180 540

POLYGONS and AREA Angles in Polygons MENU In a quadrilateral, what is the sum of the interior angles? In a hexagon, what is the sum of the interior angles? In a decagon, what is the sum of the interior angles?

POLYGONS and AREA 4. 70 x 150 90 5. Angles in Polygons z 100 110 105
MENU 4. 70 x 150 90 5. z 100 110 105 120

POLYGONS and AREA 6. 7. Angles in Polygons
MENU All the angles in the given shape are equal. Find X & Y 6. Sum of the interior angles Each of the interior angles 120 60 Each exterior angle All the angles in the given shape are equal. Find X & Y 7. Sum of the interior angles Each of the interior angles Each exterior angle 45 135

MENU 8. In a regular, convex 12-gon, what is the measure of each interior angle? 9. In a regular, convex 12-gon, what is the measure of each exterior angle? Each interior angle is 150… So each exterior angle is 30.

EXTERIOR ANGLES POLYGONS and AREA Angles in Polygons
MENU EXTERIOR ANGLES What is an EXTERIOR angle? THEOREM: The Sum of the EXTERIOR angles of a convex polygon is 3600 That’s what you get when you extend all the sides in the same direction.

EXTERIOR ANGLES POLYGONS and AREA Angles in Polygons
MENU EXTERIOR ANGLES THEOREM: The Sum of the EXTERIOR angles of a convex polygon is 3600 What is the sum of the exterior angles of an octagon? 360 What is the sum of the exterior angles of a pentagon? 360 What is the sum of the exterior angles of a decagon? 360

POLYGONS and AREA Angles in Polygons 13. 90 14. 110 X 68 63 80 117 X
MENU 13. 90 14. 110 X 68 63 80 117 X X + 63 = 360 X = 360 246 + X = 360 301 + X = 360 X = 133 X = 59

MENU 15. What is the measure of an exterior angle of a REGULAR octogon? 16. What is the measure of an exterior angle of a REGULAR decagon? 17. What is the measure of an exterior angle of a REGULAR 36-gon? 18. What is the measure of an exterior angle of a REGULAR 100-gon?

19. FIND X: POLYGONS and AREA Angles in Polygons
MENU 19. FIND X: 3x-10+2x+2x+5+x+15+2x+10 = 360 10x+20 = 360 2x 10x = 340 3x-10 x = 34 2x+5 x+15 2x+10

20. The picture shows a REGULAR HEXAGON. Find X:
POLYGONS and AREA Angles in Polygons MENU 20. The picture shows a REGULAR HEXAGON. Find X: X

POLYGONS and AREA Angles in Polygons MENU 21. A regular polygon with an unknown number of sides has exterior angles measuring How many sides does it have? X X

AREA of RECTANGLES POLYGONS and AREA Area of Squares and Rectangles
MENU AREA of RECTANGLES

Area is a measure of “flat space”.
POLYGONS and AREA Area of Squares and Rectangles MENU Area is a measure of “flat space”. If you wanted to cover a floor with 1ft by 1ft tiles, the area of the floor is the number of tiles it takes to cover the floor.

POLYGONS and AREA Here is a small room: It is 10 feet long
Area of Squares and Rectangles MENU Here is a small room: It is 10 feet long 1 and 6 feet wide 6 1 To cover the room with 1x1 tiles . . . 10 10 across by 6 deep

POLYGONS and AREA You need 10 of them across And 6 deep 6 10
Area of Squares and Rectangles MENU You need 10 of them across And 6 deep 6 10

POLYGONS and AREA For each of the 10 across, there are 6 deep
Area of Squares and Rectangles MENU For each of the 10 across, there are 6 deep How many total tiles? 6 6 x 10 = 60 This is where base x height comes from 10

Because area is measured by how many “squares” fit in a polygon…
POLYGONS and AREA Area of Squares and Rectangles MENU Because area is measured by how many “squares” fit in a polygon… 6 ft We call the units in area “SQUARE UNITS” 10 ft Area is “60 square feet”.

H B POLYGONS and AREA Area of Squares and Rectangles
MENU Formula for the area of a rectangle: B H *Base and height always make a right angle.

POLYGONS and AREA Area of Squares and Rectangles 35in 70in 14m
MENU Find the AREA and PERIMETER of each of the following: 1. 2. 35in 70in 14m

POLYGONS and AREA Area of Squares and Rectangles
MENU 3. Find the area of the shape shown. To do this, find the area of the big blue piece (pretend it doesn’t have a hole in it) 12 Then find the area of the “cutout” 3 3 Finally, subtract them. 9

MENU 4. Find the AREA and PERIMETER of the shape shown. Perimeter is easy, just add up the sides. The only trick is that you have to make sure you have all the sides.

MENU 4. Find the AREA and PERIMETER of the shape shown. To find the area, cut the shape into parts you can work with

POLYGONS and AREA Area of Triangles MENU AREA of TRIANGLES

POLYGONS and AREA Area of Triangles
MENU Each of these triangles is HALF the area of the original rectangle. Find the area of the rectangle shown: 4m 10m 4m 10m 4m 10m 4m 10m 20m2 20m2 Area = BxH =4x10 =40m2

MENU What if it is not a right triangle?

POLYGONS and AREA Area of Triangles No matter the type of triangle…
MENU No matter the type of triangle… …It is still HALF of a RECTANGLE. h b

MENU Formula for finding the area of a triangle: h h h b b b *Base and height always make a right angle.

MENU Find the AREA and PERIMETER for each of the following triangles: EXAMPLE #1 5m 3m To find the area, we need base and height 4m Remember the base and height make a right angle with each other. For perimeter, just add all the sides:

MENU Find the AREA and PERIMETER for each of the following triangles: 17ft EXAMPLE #2 8ft 15ft For perimeter, we need to know that 3rd side. We can find it using the PYTHAGOREAN THEOREM

MENU Find the AREA and PERIMETER for each of the following triangles: EXAMPLE #3 10 10 8 12

Find the AREA and PERIMETER for each of the following triangles
POLYGONS and AREA Area of Triangles MENU STUDENT PROBLEMS Find the AREA and PERIMETER for each of the following triangles 10ft 20m 16m 24ft 12m

POLYGONS and AREA Area of Triangles Find the AREA for this triangle 12
MENU Find the AREA for this triangle EXAMPLE #4 This is an easy one… …Sometimes the height is outside the triangle… 12 5 7 …but that doesn’t change anything. 8

POLYGONS and AREA Area of Triangles Find the AREA for this triangle 26
MENU Find the AREA for this triangle We need to find the height. EXAMPLE #5 This triangle is isosceles. 26 26 That means the height to the base bisects the base. h We will have to use the PYTHAGOREAN THEOREM to find the height. 24 48 24

Find the AREA and PERIMETER for each of the following triangles
POLYGONS and AREA Area of Triangles MENU STUDENT PROBLEMS Find the AREA and PERIMETER for each of the following triangles 9 4 5 17in 17in 3 6 16in

138in2 POLYGONS and AREA Area of Triangles
MENU Find the area of this shape: EXAMPLE #6 13in 5in 138in2 30in2 12in 9in 14in 108in2 12in

Find the AREA of the following shape.
POLYGONS and AREA Area of Triangles MENU STUDENT PROBLEMS Find the AREA of the following shape. 9in 135 17in 15 15in 60 8 17in

MENU These 2 triangles are similar, with a scale factor of If the area of the big one is 50cm2 then what is the area of the small one? EXAMPLE #7 12cm 20cm 3cm 5cm

MENU These 2 triangles are similar, with a scale factor of If the area of the big one is 100cm2 then what is the area of the small one? EXAMPLE #7 X Y

MENU These 2 triangles are similar, with a scale factor of If the area of the big one is 100cm2 then what is the area of the small one? X Y

MENU Finding the AREA of SIMILAR POLYGONS If 2 polygons are similar, the ratio of their areas is the square of the scale factor Area = 36m2 These 2 polygons are similar. The scale factor is If the area of the big one is 36, find the area of the other.

POLYGONS and AREA Area of Triangles A U Z W E B Y X C D
MENU STUDENT PROBLEMS A U 8 6 Z W E B A: 54m2 Y X C D First, find the scale factor Multiply the area by the scale factor twice

MENU Find the area of this regular pentagon EXAMPLE #8 10 8 24 6 12

MENU Find the area of this regular pentagon EXAMPLE #8 24 24 Area = 24x10 24 24 24 24 =240u2 24 24 24 24 10 8 24 6 12

Find the AREA of this regular octogon:
POLYGONS and AREA Area of Triangles MENU STUDENT PROBLEMS Find the AREA of this regular octogon: Area of 1 triangle: 30 # of triangles: 16 Area: 30x16 =480m2 12m 5 10m

Area of PARALLELOGRAMS
POLYGONS and AREA Area of Parallelograms MENU Area of PARALLELOGRAMS

POLYGONS and AREA Area of Parallelograms
MENU To calculate the area of a parallelogram… Just Multiply base and height H B Area of a Parallelogram h b *Base and height make a right angle.

POLYGONS and AREA Area of Parallelograms MENU Area of a RHOMBUS

MENU Calculating the area of a rhombus can be done the same as a parallelogram… OR you can use the diagonals d1 d2

This rectangle has an area of
POLYGONS and AREA Area of Parallelograms MENU Calculating the area of a rhombus can be done the same as a parallelogram… OR you can use the diagonals This rectangle has an area of A = d1 x d2 d1 So each Rhombus is half that. d2

POLYGONS and AREA Area of Parallelograms d1 d2
MENU Area of a Rhombus d1 d2 *the diagonals always make a right angle. It does NOT matter which diagonal is which. Remember the diagonal goes all the way across the shape. You will frequently be given only half of a diagonal.

MENU Find the area of each Parallelogram #1 #2 20ft 10ft 8ft 22cm 16cm 7cm

POLYGONS and AREA Area of Parallelograms Find the area of each Rhombus
MENU Find the area of each Rhombus #3 #4 #5 4m 7 3m 12 12 20ft 7 10ft

POLYGONS and AREA Area of Trapezoids MENU Area of a Trapezoid

POLYGONS and AREA Area of Trapezoids This is a trapezoid:
MENU This is a trapezoid: It has 1 set of parallel sides. Base2 Leg Leg Base1 The MIDSEGMENT … Joins the midpoints of the legs Has a length that is the average of the bases

POLYGONS and AREA Area of Trapezoids Base2 height: midsegment: Base1
MENU Like most shapes, the area of a trapezoid is based on a rectangle Base2 height: midsegment: Base1

POLYGONS and AREA Area of Trapezoids MENU Area of a Trapezoid

POLYGONS and AREA Area of Trapezoids
MENU Find the area of each of the following trapezoids: #1 #2 #3 10 6m 6mi 13mi 12mi 15 7m 9 11mi 8m 8

144 + 242 = 386u2 POLYGONS and AREA Area of Trapezoids
MENU Find the area of this shape: 10 = 144 9 386u2 22 20 11 242 22

POLYGONS and AREA Area of Trapezoids MENU Find the area:

POLYGONS and AREA Area of Trapezoids MENU Find X:

POLYGONS and AREA Circumference and Area of Circles MENU

Radius Diameter Chord Secant Tangent Point of Tangency
POLYGONS and AREA Circumference and Area of Circles MENU Radius Diameter Chord Secant Tangent Point of Tangency Distance from center to edge of a circle Distance from edge to edge of a circle through the center Any line segment that goes from edge to edge in a circle Any line that passes through a circle A line that touches the circle at exactly 1 point The point where a circle an tangent touch

Radius Diameter Chord Secant Tangent Point of Tangency
POLYGONS and AREA Circumference and Area of Circles MENU J F Radius Diameter Chord Secant Tangent Point of Tangency A C D B G H E

POLYGONS and AREA Radius Diameter Chord Secant Tangent
Circumference and Area of Circles MENU J F Radius Diameter Chord Secant Tangent Point of Tangency A C D B G H E

POLYGONS and AREA Circumference and Area of Circles
MENU For a circle, the formulas for area and perimeter are different, because there are no sides and there is no base or height.

POLYGONS and AREA Circumference and Area of Circles Radius:
MENU Radius: The distance from the center of a circle to the edge 10 5 Diameter: The distance from edge to edge of a circle, passing through the center Circumference: The distance around the outside of a circle

POLYGONS and AREA Circumference and Area of Circles What is pi
MENU What is pi Pi is what you get if you divide the distance around the outside of any circle by that circles diameter 10

POLYGONS and AREA Circumference and Area of Circles Area of a circle:
MENU Area of a circle: 5 Circumference of a circle:

MENU Find the area and circumference: AREA CIRCUMFERENCE 8

MENU Find the area and circumference: AREA CIRCUMFERENCE 12 6 Notice that this is just the diameter times pi.

POLYGONS and AREA FINDING ARCLENGTH Circumference and Area of Circles
MENU FINDING ARCLENGTH Here is a slice of pizza. 6 It has a 6 inch radius, And we cut a 60 degree Slice out of it. 60 How much crust do you have ? But you don’t have the whole Pizza, you just have 60 degrees The crust of the whole pizza: (circumference)

POLYGONS and AREA To find ARCLENGTH Circumference and Area of Circles
MENU To find ARCLENGTH 120 8

MENU Find the arclength. 5 70

POLYGONS and AREA Find the area of the SECTOR
Circumference and Area of Circles MENU Find the area of the SECTOR Area of the whole circle: Area of a sector or “slice”:

POLYGONS and AREA Find the area of the SECTOR
Circumference and Area of Circles MENU Find the area of the SECTOR

POLYGONS and AREA OPENERS MENU A B C D E F G H I J K L M N O P Q R S T
U V W X Y Z

POLYGONS and AREA OPENERS A MENU

POLYGONS and AREA OPENERS B MENU

POLYGONS and AREA OPENERS C MENU

POLYGONS and AREA OPENERS D MENU

POLYGONS and AREA OPENERS E MENU

POLYGONS and AREA OPENERS F MENU

POLYGONS and AREA OPENERS G MENU

Can you do this WITHOUT a CALCULATOR?
POLYGONS and AREA OPENERS H MENU Can you do this WITHOUT a CALCULATOR? In your job as a cashier, a customer gives you a $20 bill to pay for a can of coffee that costs $3.84. How much change should you give back? a) $ b) $ c) $ d) $ e) $17.16

POLYGONS and AREA OPENERS I MENU How much time is there between 7:35 a.m. and 5:25 p.m.? a) 8 hours and 50 minutes b) 9 hours and 10 minutes c) 9 hours and 50 minutes d) 10 hours e) 10 hours and 50 minutes

POLYGONS and AREA OPENERS J MENU

POLYGONS and AREA OPENERS K MENU

POLYGONS and AREA OPENERS L MENU

POLYGONS and AREA OPENERS M MENU

POLYGONS and AREA OPENERS N MENU

POLYGONS and AREA OPENERS O MENU

POLYGONS and AREA OPENERS P MENU

POLYGONS and AREA OPENERS Q MENU

POLYGONS and AREA OPENERS R MENU

POLYGONS and AREA OPENERS S MENU

POLYGONS and AREA Reviews Find the Area (assorted)
MENU Find the Area (assorted) Compound Polygon (5-Questions) Jeopardy Review Review for Quiz Review for Quiz Review Harder Problems

POLYGONS and AREA Reviews Find the area of each shape: MENU 3 answers
1 2 3 4 154 24 6 12 5 6 7 8 3 69 28 31 126 9 10 11 12 29 52 answers

POLYGONS and AREA Reviews MENU

POLYGONS and AREA Reviews MENU Review for quiz 8.1-8.2
VOCABULARY: Equilateral, Equiangular, Regular, Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon, Convex, Concave, Interior, Exterior

POLYGONS and AREA Classifying Polygons MENU
Identify as CONVEX or CONCAVE

MENU What is the sum of the interior angles of a decagon? What is the sum of the exterior angles of a pentagon? x4

POLYGONS and AREA Classifying Polygons Find X 110 112 130 138 113 X x4
MENU Find X 90 110 112 68 63 130 138 80 X 113 X X + 63 = 360 X =720 301 + X = 360 X=117 X = 59 x4

MENU What is the measure of each interior angle of a regular 30-gon? What is the measure of each exterior angle of a regular hexagon? A regular polygon has exterior angles measuring How many sides does it have? x4

POLYGONS and AREA Review MENU

POLYGONS and AREA Review HARDER PROBLEMS
MENU #1 Find the area of the RED region

POLYGONS and AREA Review HARDER PROBLEMS #2 Find the measure of X.
MENU #2 Find the measure of X.

POLYGONS and AREA Review HARDER PROBLEMS MENU #3 Find the area:

POLYGONS and AREA Review HARDER PROBLEMS MENU #4

MENU #5 Find the measure of the area of the sector.

MENU #6 The area of the square shown is 144 in2. Find the length of the sides.

MENU #7 Find X if the area of the trapezoid is 48in2: 4in X 12in

MENU #8 Find the area of the RED region:

POLYGONS and AREA Classifying Polygons Angles in Polygons

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