Proportions with Perimeter, Area, and Volume Chapter 11.5

[You will need graph paper] Objective To discover the relationship between the perimeters, areas and volumes of similar figures [You will need graph paper]

On Graph Paper Draw rectangle ABCD with length and width of 16 and 12 Draw rectangle EFGH with length and width of 12 and 9 Write a similarity statement for the two rectangles

Yours should look something like these F E D C B A 12 16 9 ABCD~EFGH

Remember, this is called the Ratios H G F E D C B A 12 16 9 Compare the larger rectangle to the smaller rectangle. Write the ratio of any two corresponding sides. Remember, this is called the LINEAR Ratio AKA: Similarity Ratio Scale Factor

Ratios H G F E D C B A 12 16 9 P=56 P=42 Calculate the perimeter of each rectangle What is the ratio of the larger perimeter to the smaller?

Ratios Linear Similarity Conjecture H G F E D C B A 12 16 9 P=56 P=42 Since the perimeter is a LINEAR measurement, it is in the same LINEAR ratio. Linear Similarity Conjecture The ratios of any corresponding linear measures of similar figures are equal to the ratio of corresponding sides

Linear Measurements

Linear ratio MUST be in simplest form Linear Measurements Perimeter Radius Length Circumference Width Height Diameter Linear ratio MUST be in simplest form Now back to the rectangles…

Area Calculate the Area of each rectangle What is the ratio of the larger to smaller areas? H G F E D C B A 12 16 9 A=192 A=108

Area How does this ratio compare to the linear ratio? (The linear ratio was ) H G F E D C B A 12 16 9 A=192 A=108

Area Draw another set of similar rectangles on your paper and see if your theory works again (try starting with a 5x7 rectangle and choosing a scale factor to make a second rectangle)

Linear ratio MUST be in simplest form Area Proportional Areas Conjecture If corresponding sides of two similar polygons or the radii of two circles compare in the ratio , then their areas compare in the ratio Linear ratio MUST be in simplest form

Check This Out (?) NCTM Applet for Perimeter and Area

Area Examples Linear (L): Area (A): The ratio of the corresponding midsegments of two similar trapezoids is 4:5. What is the ratio of their areas? Linear (L): Area (A):

Find the Linear and Area ratios Area Examples Find the Linear and Area ratios L: A:

Area Examples Fun Fact: ALL circles are similar! L: A: A=560π cm What is the area of circle N (in terms of π)? L: A: q m P N A=560π cm

Volume Consider these rectangular prisms Are all of their corresponding linear measures proportional? What is the linear ratio? What is the area ratio? L: A: FIX ME!!! 1.5 1 4.5 3 2 3

Volume Find the volume of each prism What is the ratio of the volumes? 1.5 V=20.25 1 V=6 4.5 3 2 3

Volume L: A: and the ratio of VOLUMES is Is there a relationship? 1.5 V=20.25 1 V=6 4.5 3 2 3

Volume Proportional Volumes Conjecture If corresponding edges (or radii, height, etc.) of two similar solids compare in the ratio , then their areas compare in the ratio

Check This Out (?) NCTM Applet for Volume of Similar Solids

Volume Examples L: A: V: The corresponding heights of two similar cylinders is 2:5. What are the Linear, Area and Volume ratios? L: A: V: FIND ALL ratios!! 5 2

Volume Examples X Y L: A: VX = 35.1ft3 V: 9ft k Triangular prisms X and Y are similar The linear ratio is ¾. Find the area and volume ratios. L: A: VX = 35.1ft3 X Y V: 9ft k

Volume Examples X Y L: A: V: VX = 35.1ft3 9ft k Triangular prisms X and Y are similar The linear ratio is ¾. Find K L: A: V: VX = 35.1ft3 X Y 9ft k

Volume Examples X Y L: A: V: VX = 35.1ft3 9ft k Triangular prisms X and Y are similar The linear ratio is ¾. Find volume of prism Y L: A: V: VX = 35.1ft3 X Y 9ft k

You can’t jump between area and volume without going through linear The Ratios Linear Area Volume You can’t jump between area and volume without going through linear