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.

Area A measurement of the region covered by a geometric figure and its interior.

Theorem The area of a rectangle is the product of the base and height. b h Area = b x h

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

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

Postulate The area of a square is the length of the side squared. s s Area = s 2

If two figures are congruent, then they have the same area. Postulate A B If triangle A is congruent to triangle B, then area A = area B.

Find the area

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

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 b h

White Board Practice b12m9cm y-2 h3m6cm y A36m 2 54 cm 2 y 2 – 2y b h

Group Practice Find the area of the figure. Consecutive sides are perpendicular

Group Practice Find the area of the figure. Consecutive sides are perpendicular A = 114

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

Tons of formulas to memorize So don’t memorize them Understand them !

Refresh my memory… What is the area of a rectangle ?

Refresh my memory… what is the height in a rectangle? Altitude to a base is any segment perpendicular to the line containing the base from any point on the opposite side.

h

h

h

h Cut Slide over and tape

h

Do we have any leftover paper?

So this means the area must be the same

Theorem The area of a parallelogram is the product of the base times the height to that base. b h Area = b x h

But Wait….

What do we have ? X 2

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

Theorem The area of a rhombus equals half the product of the diagonals. d1d1 d2d2

Remote Time

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

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

White Board Practice Find the area of the figure º

White Board Practice Find the area of the figure º

White Board Practice Find the area of the figure 5 5 6

White Board Practice Find the area of the figure 5 5 6

White Board Practice Find the area of the figure

White Board Practice Find the area of the figure

White Board Practice Find the area of the figure

White Board Practice Find the area of the figure

White Board Practice Find the area of the figure

White Board Practice Find the area of the figure

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 height median

Median Remember the median is the segment that connects the midpoints of the legs of a trapezoid. Length of median = ½ (b 1 +b 2 ) b1b1 b2b2 median

Height The height of the trapezoid is the segment that is perpendicular to the bases of the trapezoid b1b1 h b2b2

Theorem The area of a trapezoid equals half the product of the height and the sum of the bases. b1b1 h b2b2 Sketch

White Board Practice Find the area of the trapezoid and the length of the median

White Board Practice Find the area of the trapezoid and the length of the median A = 50 Median = 10

White Board Practice Find the area of the trapezoid and the length of the median 10

White Board Practice Find the area of the trapezoid and the length of the median 10 A = 54 Median = 9

White Board Practice 3. Find the area of the trapezoid and the length of the median

White Board Practice 3. Find the area of the trapezoid and the length of the median A = 138 Median = 11.5

Group Practice A trapezoid has an area of 75 cm 2 and a height of 5 cm. How long is the median?

Group Practice A trapezoid has an area of 75 cm 2 and a height of 5 cm. How long is the median? Median = 5 cm

Group Practice Find the area of the trapezoid º

Group Practice Find the area of the trapezoid º Area =

Group Practice Find the area of the trapezoid 45º 4

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

Group Practice Find the area of the trapezoid º

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 All angles congruent

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

Radius of a regular polygon center 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

Apothem of a regular polygon apothem the perpendicular distance from the center to a side of the polygon

Regular Polygon Review side center radius apothem central angle Perimeter = sum of sides

Polygon Review Continued SidesName1 interior  1 Central  3Triangle Square Pentagon Hexagon Septagon Octagon nn-gon(n-2) n n

Theorem The area of a regular polygon is half the product of the apothem and the perimeter. a s r P = 8s

RAPA R adius A pothem P erimeter A rea a s r

Radius, Apothem, Perimeter 1.Find the central angle 360 n

Radius, Apothem, Perimeter 2. Divide the isosceles triangle into two congruent right triangles

Radius, Apothem, Perimeter r a 3. Find the missing pieces x

Radius, Apothem, Perimeter Think Think Thing SOHCAHTOA

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

r a p A r a x 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap

r a p A 12 r a x 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap

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

r a p A 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap r a x

r a p A 21 1.Central angle 2.½ of central angle SOHCAHTOA A = ½ ap r a x

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

r a p A ?? Doesn’t work! Why?

r a p A ?? Doesn’t work! Why?

Circumference The distance around the outside of a circle.

Experiment 1.Select 5 circular objects 2.Using a piece of string measure around the outside of one of the circles. 3.Using a ruler measure the piece of string to the nearest mm. 4.Using a ruler measure the diamter to the nearest mm. 5.Record in the table.

Experiment 6. Make a ratio of the Circumference. Diameter 7. Give the ratio in decimal form to the nearest hundreth.

Experiment Circle Number Circumference (nearest mm) Diameter (nearest mm) Ratio of Circumference/Diameter (as a decimal)

What do you think? 1. How does the measurement of the circumference compare to the measurement of the diameter?

2. Were there any differences in results? If so, what were they?

3. Did you recognize a pattern? Were you able to verify a pattern?

 Greek Letter Pi (pronounced “pie”) Pi is the ratio of the circumference of a circle to the diamter. Ratio is constant for all circles Irrational number Common approximations –3.14 – –22/7

Circumference The distance around the outside of a circle. r

Area B The area of a circle is the product of pi times the square of the radius. r

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

Remember Circumference C = 2  r

Arc Length The length of an arc is the product of the circumference of the circle and the ratio of the circle that the arc represents. O B C x r

Remember Area ?

Sector Area The area of a sector is the product of the area of the circle and the ratio of the circle that the sector of the circle represents. A B C x r

White Board Practice

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

Comparing Areas of Triangles

Two triangles with equal heights 4 4

4 4

Ratio of their areas

Ratio of areas = ?

If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.

Two triangles with equal bases

Ratio of Areas

Ratio of Areas = ?

If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.

If two triangles are similar, then the ratio of their areas equals the square of their scale factor.

Theorem If the scale factor of two similar triangles is a:b, then 1.)the ratio of their perimeters is a:b 2.)the ratio of their areas is a 2 :b 2.

White Board Practice Find the ratio of the areas of  ABC:  ADB A B C D

White Board Practice Find the ratio of the areas of  ABD:  BCD A B C D

Remember Scale Factor a:b Ratio of perimeters a:b Ratio of areas a 2 :b 2

Remote Time True or False

T or F If two quadrilaterals are similar, then their areas must be in the same ratio as the square of the ratio of their perimeters

T or F If the ratio of the areas of two equilateral triangles is 1:3, then the ratio of the perimeters is 1:

T or F If the ratio of the perimeters of two rectangles is 4:7, then the ratio of their areas must be 16:49

T or F If the ratio of the areas of two squares is 3:2, then the ratio of their sides must be

11-8: Geometric Probability Solve problems aboutGeometric probability.

Event: A possible outcome in a random experiment.

Sample Space The number of all possible outcomes in a random experiment.

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

Geometric Probability The area of the event divided by the area of the sample space. The length of an event divided by the length of the sample space.

Homework Set 11.8 Pg 463 (2-10 even Pg 465 (1-10)