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 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…. Notation 1 unit - 2cm - 3in 2 units 2 3 cm 2 – 10 in 2 4 units 3 8 cm 3
Area A measurement of the region covered by a geometric figure and its interior. What types of jobs use area everyday?
If two figures are congruent, then they have the same area. Area Congruence Postulate A B 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. A B C Area of figure = Area A + Area B + Area 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… 1.What is the area formula for this shape? 2.What part of the formula do I already have? 3.What part do I need to find? 4.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. s s Area = s 2 What’s the are of a square with.. side length of 4? perimeter of 12 ?
Theorem The area of a rectangle is the product of the base and height. b h Area = b x h Using the variables shown on the diagram create an equation that would represent the perimeter of the figure.
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 b12m9cm y-2 h3m y A54 cm 2 P b h
Group Practice b12m9cm y-2 h3m6cm y A36m 2 54 cm 2 y 2 – 2y P 30m 4y-4 b h
Find the area of the rectangle 5 3 AREA = 12
Group Practice Find the area of the figure. Consecutive sides are perpendicular A = 114 units 2
Finding area? Ask these questions… 1.What is the area formula for this shape? 2.What part of the formula do I already have? 3.What part do I need to find? 4.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? b h
Theorem The area of a parallelogram is the product of the base times the height to that base. b h Area = b x h **This right triangle is key to helping solve!!
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. b h A = bh 2
Partners How would you label the base and height of these triangles?
Theorem The area of a rhombus equals half the product of the diagonals. d1d1 d2d2 **WHAT DO YOU SEE WITHIN THE DIAGRAM? A = d 1 ∙ d 2 _________ 2
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… 1.What is the area formula for this shape? 2.What part of the formula do I already have? 3.What part do I need to find? 4.How can I use a right triangle to find the missing part?
White Board Practice Just talk about this one 5 5 6
White Board Practice Find the area of the figure º
White Board Practice Find the area of the figure Just talk about this one
White Board Practice Find the area of the figure 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 b1b1 h b2b2 Partners: Why is the height perpendicular to both bases? How do we measure height for a trap?
Theorem The area of a trapezoid equals half the product of the height and the sum of the bases. b1b1 h b2b2 demo
Labeling Height for Isosceles Trap Always label 2 heights when dealing with an isosceles trap
White Board Practice Find the area of the trapezoid A = 50 **talk**
White Board Practice 3. Find the area of the trapezoid A = 138 *talk*
Finding area? Ask these questions… 1.What is the area formula for this shape? 2.What part of the formula do I already have? 3.What part do I need to find? 4.How can I use a right triangle to find the missing part?
Group Practice Find the area of the trapezoid º Area =
Group Practice Find the area of the trapezoid Area = 45º 4
Group Practice Find the area of the trapezoid Area = º
11.4 Areas of Regular Polygons Objectives Determine the area of a regular polygon.
Regular Polygon Review side All sides congruent (n-2) 180 n All angles congruent
Circles and Regular Polygons Read Pg. 440 and 441 –Start at 2 nd 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 center is the center of the circumscribed circle
Radius of a regular polygon radius is the radius of the circumscribed circle is the distance from the center to a vertex
Central angle of a regular polygon Central angle Is an angle formed by two radii drawn to consecutive vertices How many central angles does this regular pentagon have? 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 apothem the perpendicular distance from the center to a side of the polygon How many apothems does this regular pentagon have? How many apothems does a regular triangle have?
Regular Polygon Review side center radius apothem central angle Perimeter = sum of sides **What do you think the apothem does to the central angle?
Theorem The area of a regular polygon is half the product of the apothem and the perimeter. a s r s = length of side p = 8s What does each letter represent in the diagram? A = ap 2
RAPA R adius A pothem P erimeter A rea a s r This right triangle is the key to finding each of these parts.
Radius, Apothem, Perimeter 1.Find the central angle 360 n
Radius, Apothem, Perimeter 2.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 r a 3.Find the missing pieces What does ‘x’ represent? x
Radius, Apothem, Perimeter Think Think Think SOHCAHTOA
r a p A 8 r a x 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap
r a p A 84 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap r a x IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE?
r a p A r a x 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap
r a p A r a x A = ½ ap IS THERE ANOTHER AREA FORMULA FOR THIS SHAPE?
r a p A 8 r a x 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap
r a p A Central angle 2.½ of central angle SOHCAHTOA A = ½ ap r a x
r a p A 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap r a x
r a p A 6 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap 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 = CdCd
Circumference The distance around the outside of a circle. r C = circumference r = radius d = diameter d r C = ∏ d C = ∏ 2r
Area B The area of a circle is the product of pi times the square of the radius. r For both formulas always leave answers in
WHITEBOARDS rdCA ∏ 100∏ 18∏ *put answers in terms of pi
Quiz review - Set up these diagrams 1.A square with side 2√3 2.A rectangle with base √4 and diagonal √5 3.A parallelogram with sides 6 and 10 and a 45◦ angle 4. A rhombus with side 10 and a diagonal 12 5.An isosceles trapezoid with bases of 2 and 6 and base angles that measure 45 ◦ 6.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. r A B
Warm - up 1.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? B A C 60
The distance around the outside of a circle. length Finding the total length Remember Circumference B C r x◦
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? O x◦ Degree measure of arc LENGTH OF ARC Circumference of circle
Example If r = 6, what is the length of CB? O B C 60◦ Measure of CB = 60◦ 60 = (2 ∙ 6) = 22
Remember Area B C Sector of a circle 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? O x◦ Degree measure of arc AREA OF SECTOR Area of circle
Example If r = 6, what is the area of sector COB? O B C 60◦ Measure of CB = 60◦ 60 = ( ∙ 6 2 ) = 66
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 (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 O B A 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 abab 1 : 2 3 : 5a : b 1 to 23 to 5a to b
Solving a Proportion First, cross-multiply Next, divide by 5
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: **What have we used scale factor for in past chapters?
Theorem If the scale factor of two similar figures is a:b, then… 1.the ratio of their perimeters is a:b 2.the ratio of their areas is a 2 :b 2. Area = Scale Factor- 7: 3 Ratio of P – 7: 3 Ratio of A – 49 :9 ~
WHITEBOARDS OPEN BOOK TO PG. 458 (classroom exercises) ANSWER #4 –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 a.No b. ADE ~ ABC c. 4: 25 d. 4:21
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 a 2 :b 2
11-8: Geometric Probability Solve problems about geometric probability
Read Pg. 461 Solving Geometric Problems using 2 principles 1.Probability of a point landing on a certain part of a line (length) 2.Probability of a point landing in a specific region of an area (area)
Sample Space The number of all possible outcomes in a random experiment. 1.Total length of the line 2.Total area
Event: A possible outcome in a random experiment. 1.Specific segment of the line 2.Specific region of an area
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 1.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 2.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
WHITEBOARDS OPEN BOOK TO PG. 462 (classroom exercises) ANSWER #2 –1 / 3 ANSWER # 3 –Give answer in terms of pi
WHITEBOARDS Find the ratio of the areas of WYV to XYZ –4 to 49 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/ X Z Y W V
Drawing Quiz- Set up these diagrams 1.A rectangle with base 10 and diagonal 15 2.A parallelogram with sides 6 and 10 and a 60◦ angle 3. A rhombus with side 10 and a diagonal 12 4.An equilateral triangle with a perimeter = 27 5.Sector AOB: AO = 12 and the central angle equals 50 degrees 6.Isosceles triangle with base of 10 and perimeter of 40.
Test Review Chapter Review Chapter test –4 –9 –12 –15