Chapter 11 Areas of Plane Figures Understand what is meant by the area of a polygon. Know and use the formulas for the areas of plane figures. Work geometric probability problems.

11-1: Area of Rectangles Objectives Learn and apply the area formula for a square and a rectangle.

Math Notation for Different Measurements Dimensions Notation Length (1 dimension) The length of a line is…. Area (2 dimensions) The area of a rectangle is …. Volume (3 dimensions) The volume of a cube is…. 1 unit - 2cm - 3in 2 units2 3 cm2– 10 in2 4 units3 8 cm3

Area A measurement of the region covered by a geometric figure and its interior. What types of jobs use area everyday?

Area Congruence Postulate If two figures are congruent, then they have the same area. B A If triangle A is congruent to triangle B, then area A = area B. With you partner: Why would congruent figures have the same area?

Area Addition Postulate The area of a region is the sum of the areas of its non-overlapping parts. Area of figure = Area A + Area B + Area C B A C

Base (b) Any side of a rectangle or other parallelogram can be considered to be a base.

Altitude (Height (h)) Altitude to a base is any segment perpendicular to the line containing the base from any point on the opposite side. Called Height

Finding area? Ask these questions… What is the area formula for this shape? What part of the formula do I already have? What part do I need to find? How can I use a right triangle to find the missing part?

Postulate The area of a square is the length of the side squared. Area = s2 s What’s the are of a square with.. side length of 4? perimeter of 12 ? s

Theorem The area of a rectangle is the product of the base and height. h Area = b x h Using the variables shown on the diagram create an equation that would represent the perimeter of the figure. b

Remote Time Classify each statement as True or False

Question 1 If two figures have the same areas, then they must be congruent.

Question 2 If two figures have the same perimeter, then they must have the same area.

Question 3 If two figures are congruent, then they must have the same area.

Question 4 Every square is a rectangle.

Question 5 Every rectangle is a square.

Question 6 The base of a rectangle can be any side of the rectangle.

White Board Practice h b b 12m 9cm y-2 h 3m y A 54 cm2 P

Group Practice b 12m 9cm y-2 h 3m 6cm y A 36m2 54 cm2 y2 – 2y P 30m

Find the area of the rectangle 5 3 AREA = 12

A = 114 units2 Group Practice Find the area of the figure. Consecutive sides are perpendicular. 3 2 4 5 5 A = 114 units2 6

Finding area? Ask these questions… What is the area formula for this shape? What part of the formula do I already have? What part do I need to find? How can I use a right triangle to find the missing part?

11-2: Areas of Parallelograms, Triangles, and Rhombuses Objectives Determine and apply the area formula for a parallelogram, triangle and rhombus.

Base (b) and Height (h)

PARTNERS…. How do a rectangle and parallelogram relate? What could I do with this parallelogram to make it look like a rectangle? h b

Theorem The area of a parallelogram is the product of the base times the height to that base. **This right triangle is key to helping solve!! h Area = b x h b

Triangle Demo How can I take two congruent triangles and connect them to make a new shape?

Theorem The area of a triangle equals half the product of the base times the height to that base. A = bh 2 h b

Partners How would you label the base and height of these triangles?

2 A = d1∙d2 Theorem _________ The area of a rhombus equals half the product of the diagonals. A = d1∙d2 _________ 2 d1 d2 **WHAT DO YOU SEE WITHIN THE DIAGRAM?

Organization is Key Always draw the diagrams Know what parts of the formula you have and what parts you need to find Right triangles will help you find missing information

Finding area? Ask these questions… What is the area formula for this shape? What part of the formula do I already have? What part do I need to find? How can I use a right triangle to find the missing part?

White Board Practice 5 5 6 Just talk about this one

White Board Practice Find the area of the figure 6 3 3 60º 6

White Board Practice Find the area of the figure 12 5 13 Just talk about this one Just talk about this one

White Board Practice Find the area of the figure 2 5 5 2 Just talk about this one

White Board Practice Find the area of the figure Side = 5cm 1 diagonal = 8cm

White Board Practice Find the area of the figure 4 4 4

11-3: Areas of Trapezoids Objectives Define and apply the area formula for a trapezoid.

Trapezoid Review A quadrilateral with exactly one pair of parallel sides. base leg median What type of trap do we have if the legs are congruent?

Height The height of the trapezoid is the segment that is perpendicular to the bases of the trapezoid b2 How do we measure height for a trap? h Partners: Why is the height perpendicular to both bases? b1

Theorem The area of a trapezoid equals half the product of the height and the sum of the bases.   b1 h b2 demo

Labeling Height for Isosceles Trap Always label 2 heights when dealing with an isosceles trap

White Board Practice A = 50 1. Find the area of the trapezoid 7 5 13 **talk**

White Board Practice A = 138 3. Find the area of the trapezoid 13 14 9 12 *talk*

Finding area? Ask these questions… What is the area formula for this shape? What part of the formula do I already have? What part do I need to find? How can I use a right triangle to find the missing part?

Group Practice Find the area of the trapezoid 8 8 8 60º Area =

Group Practice Find the area of the trapezoid 45º 4 Area =

Group Practice Find the area of the trapezoid 12 30º 30º 30 Area =

11.4 Areas of Regular Polygons Objectives Determine the area of a regular polygon.

Regular Polygon Review All sides congruent All angles congruent (n-2) 180 n side

Circles and Regular Polygons Read Pg. 440 and 441 Start at 2nd paragraph, “Given any circle… What does it mean that we can inscribe a poly in a circle? Each vertex of the poly will be on the circle

Center of a regular polygon is the center of the circumscribed circle center

Radius of a regular polygon is the distance from the center to a vertex is the radius of the circumscribed circle radius

Central angle of a regular polygon Is an angle formed by two radii drawn to consecutive vertices How many central angles does this regular pentagon have? Central angle How many central angles does a regular octagon have?

Think – Pair – Share What connection do you see between the 360◦ of a circle and the measure of the central angle of the regular pentagon? Central angle 360 n

Apothem of a regular polygon the perpendicular distance from the center to a side of the polygon How many apothems does this regular pentagon have? apothem How many apothems does a regular triangle have?

Regular Polygon Review **What do you think the apothem does to the central angle? central angle center Perimeter = sum of sides radius apothem side

What does each letter represent in the diagram? Theorem The area of a regular polygon is half the product of the apothem and the perimeter. What does each letter represent in the diagram? s = length of side p = 8s r A = ap 2 a s

RAPA R adius A pothem P erimeter A rea r a s This right triangle is the key to finding each of these parts.

Radius, Apothem, Perimeter Find the central angle 360 n

Radius, Apothem, Perimeter Draw in the apothem… This divides the isosceles triangle into two congruent right triangles How do we know it’s an isosceles triangle?

Radius, Apothem, Perimeter x Find the missing pieces What does ‘x’ represent?

Radius, Apothem, Perimeter Think 30-60-90 Think 45-45-90 Think SOHCAHTOA

8 A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE? 8 4 A = ½ ap Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE? x

A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE? 5 40 100 A = ½ ap r a IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE? x

8 A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

8 48 A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

6 A = ½ ap r a p A Central angle ½ of central angle 45-45-90 30-60-90 SOHCAHTOA r a x

11.5 Circumference and Areas of Circles Objectives Determine the circumference and area of a circle. r

 Greek Letter Pi (pronounced “pie”) Used in the 2 main circle formulas: Circumference and Area (What are these?) Pi is the ratio of the circumference of a circle to the diameter. Ratio is constant for ALL CIRCLES Irrational number (cannot be expressed as a ratio of two integers) Common approximations 3.14 22/7

Circumference The distance around the outside of a circle. **The Circumference and the diameter have a special relationship that lead us to  = C d

Circumference C = ∏ d C = ∏ 2r C = circumference r = radius The distance around the outside of a circle. C = ∏ d C = ∏ 2r r r C = circumference r = radius d = diameter d

Area The area of a circle is the product of pi times the square of the radius. r B For both formulas always leave answers in 

WHITEBOARDS *put answers in terms of pi r d C A 15 8 26∏ 100∏ 18∏

Quiz review - Set up these diagrams A square with side 2√3 A rectangle with base √4 and diagonal √5 A parallelogram with sides 6 and 10 and a 45◦ angle A rhombus with side 10 and a diagonal 12 An isosceles trapezoid with bases of 2 and 6 and base angles that measure 45 ◦ A regular hexagon with a perimeter 72

11.6 Arc Length and Areas of Sectors Objectives Solve problems about arc length and sector and segment area. A B r

Warm - up If you had the two pizzas on the right and you were really hungry, which one would you take a slice from? Why? Same angle

Arc Measure tells us the fraction or slice represents… How much of the 360 ◦ of crust are we using from our pizza? 60 A C 60 B

Remember Circumference The distance around the outside of a circle. x◦ Finding the total length B C x◦ r

Arc Length The length of the arc is part of the circle’s circumference… the question is, what fraction of the total circumference does it represent? x◦ Circumference of circle x◦ Degree measure of arc O LENGTH OF ARC

Example If r = 6, what is the length of CB? 1 (2 ∙ 6) = 2 6 Measure of CB = 60◦ 60 = 1 360 6 B C 60◦ 1 6 (2 ∙ 6) = 2 O

Remember Area Sector of a circle B C aka – the area of the piece of pizza

Area of a Sector The area of a sector is part of the circle’s area… the question is, what fraction of the total area does it represent? x◦ Area of circle x◦ Degree measure of arc O AREA OF SECTOR

Example If r = 6, what is the area of sector COB? 1 6 ( ∙ 62) = 6 Measure of CB = 60◦ 60 = 1 360 6 B C 60◦ O 1 6 ( ∙ 62) = 6

REMEMBER!!! Both arc length and the area of the sector are different with different size circles! Just think pizza

WHITEBOARDS ONE PARTNER OPEN BOOK TO PG. 453 ANSWER #2 ANSWER # 4 (classroom exercises) ANSWER #2 Length = 4 Area = 12 ANSWER # 4 Length = 6 Area = 12 ANSWER #1 (we)

WHITEBOARDS Find the area of the shaded region 25∏ - 50 B 10 A O 10

11-7 Ratios of Areas Objectives Solve problems about the ratios of areas of geometric figures.

Ratio A ratio of one number to another is the quotient when the first number is divided by the second. A comparison between numbers There are 3 different ways to express a ratio This just defines a ratio. Can any of these ratios be simplified? What could the extended ratio be used to do? 3 5 1 2 a b 3 : 5 a : b 1 : 2 1 to 2 3 to 5 a to b 94

Solving a Proportion First, cross-multiply Next, divide by 5 This shows how to solve a proportion. A proportion can be solved if it is expressed with a single variable. 95

**What have we used scale factor for in past chapters? The Scale Factor If two polygons are similar, then they have a scale factor The reduced ratio between any pair of corresponding sides or the perimeters. 12:3  scale factor of 4:1 **What have we used scale factor for in past chapters? 12 Work several examples of how to find the scale factor, and how to use it to find the unknown parts. 3 96

Theorem If the scale factor of two similar figures is a:b, then… the ratio of their perimeters is a:b the ratio of their areas is a2:b2. Scale Factor- 7: 3 Ratio of P – 7: 3 Ratio of A – 49 :9 Area = 27 ~ 7 3

WHITEBOARDS OPEN BOOK TO PG. 458 (classroom exercises) ANSWER # 13 Ratio of P – 1:3 Ratio of A – 1:9 If the smaller figure has an area of 3 what is the area of the larger shape? ANSWER # 10 Scale factor – 4:7 Ratio of P – 4:7 ANSWER # 13 No b. ADE ~ ABC c. 4: 25 d. 4:21 98

WHITEBOARDS The areas of two similar triangles are 36 and 81. The perimeter of the smaller triangle is 12. Find the perimeter of the bigger triangle. 36/81 = 4/9  2/3 is the scale factor 2/3 = 12/x  x = 18

Remember Scale Factor a:b Ratio of perimeters a:b Ratio of areas a2:b2

11-8: Geometric Probability Solve problems about geometric probability

Read Pg. 461 Solving Geometric Problems using 2 principles Probability of a point landing on a certain part of a line (length) Probability of a point landing in a specific region of an area (area)

The number of all possible outcomes in a random experiment. Sample Space The number of all possible outcomes in a random experiment. Total length of the line Total area

A possible outcome in a random experiment. Event: A possible outcome in a random experiment. Specific segment of the line Specific region of an area

The calculation of the possible outcomes in a random experiment Probability The calculation of the possible outcomes in a random experiment

For example: When I pull a popsicle stick from the cup, what is the chance I pull your name?

Geometric Probability The length of an event divided by the length of the sample space. In a 10 minute cycle a bus pulls up to a hotel and waits for 2 minutes while passengers get on and off. Then the bus leaves. If a person walks out of the hotel front door at a random time, what is the probability that the bus is there?

Geometric Probability The area of an event divided by the area of the sample space. If a beginner shoots an arrow and hits the target, what is the probability that the arrow hits the red bull’s eye? 1 2 3 108

WHITEBOARDS OPEN BOOK TO PG. 462 (classroom exercises) ANSWER #2 1 / 3 ANSWER # 3 Give answer in terms of pi 109

WHITEBOARDS Y Find the ratio of the areas of WYV to XYZ Find the ratio of the areas of WYV to quad WVZX 4 to 45 Find the probability of a point from the interior of XYZ will lie in the interior of quad XWYZ 45/49 Y 2 W V 5 X Z

Drawing Quiz- Set up these diagrams A rectangle with base 10 and diagonal 15 A parallelogram with sides 6 and 10 and a 60◦ angle A rhombus with side 10 and a diagonal 12 An equilateral triangle with a perimeter = 27 Sector AOB: AO = 12 and the central angle equals 50 degrees Isosceles triangle with base of 10 and perimeter of 40.

Test Review Chapter Review 16 12 21 22 Chapter test 4 9 12 15