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Here are a few recommended readings before getting started with this lesson.
On a trip to Egypt, Emily is visiting the Great Egyptian Museum. She sees two models of Pyramids, Khafra and Menkaure, on display. She decides to buy the model of Menkaure, but now as she exits the gift shop, she has become even more curious about the volume of Khafra.
Suppose that the models of the pyramids are similar. If the scale factor of the corresponding side lengths is 1:2 and the volume of the smaller pyramid is 20 cubic centimeters, what is the volume of the larger model?
If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
Let KLMN and PQRS be similar figures, and A1 and A2 be their respective areas. The length scale factor between corresponding side lengths is ba. Here, the following conditional statement holds true.
KLMN∼PQRS⇒A2A1=(ba)2
The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.
The area of a rectangle is the product of its length and its width.
Area of KLMN | Area of PQRS |
---|---|
A1=KL⋅LM | A2=PQ⋅QR |
KL=PQ⋅ba, LM=QR⋅ba
Remove parentheses
Commutative Property of Multiplication
a⋅a=a2
Associative Property of Multiplication
Dominika is making a flyer for a concert and wants to compare the areas of the two pieces of paper. The widths of A4 and A2 papers are 21 and 42 centimeters, respectively.
If these two pieces of paper are similar and the area of A4 paper is about 624 square centimeters, find the area of the A2 paper. Round the answer to the nearest integer.
If the scale factor of two similar figures is ba, then the ratio of their areas is b2a2.
The two pieces of paper are similar and two corresponding sides measure 21 centimeters and 42 centimeters.
Therefore, the scale factor is the ratio of these corresponding sides.LHS⋅A2=RHS⋅A2
A2a⋅A2=a
b1⋅a=ba
LHS⋅4=RHS⋅4
Rearrange equation
It was just learned that if the length scale factor of two similar figures and one of the areas of the figures are known, then the unknown area can be found. Next, consider if both areas but only the side length of one figure are known. It will be possible to solve for the other similar figure's corresponding side length.
The diagram shows two similar figures. Figure A has an area of 9 square inches, and Figure B has an area of 25 square inches.
If a side length of Figure B is 2.5 inches, find the length of the corresponding side in the other shape.If the scale factor of two similar figures is ba, then the ratio of their areas is b2a2.
LHS⋅2.5=RHS⋅2.5
ca⋅b=ca⋅b
Calculate quotient
Rearrange equation
Determine the linear scale factor of the shape on the right to the shape on the left.
For similar three-dimensional figures, the volume scale factor and the length scale factor are also related.
If two figures are similar, then the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths.
Let Solid A and Solid B be similar solids and V1 and V2 be their respective volumes. The length scale factor between corresponding linear measures is ba. Given these characteristics, the following conditional statement holds true.
Solid A∼Solid B⇒V2V1=(ba)3
The statement will be proven for similar rectangular prisms, but this proof can be adapted to prove other similar solids. As shown in the diagram, let a1, a2, and a3 be the dimensions of Solid A and b1, b2, and b3 be the dimensions of Solid B.
The volume of a rectangular prism is the product of its base area and its height.
Volume of Solid A | Volume of Solid B |
---|---|
V1=a1⋅a2⋅a3 | V2=b1⋅b2⋅b3 |
Substitute expressions
Remove parentheses
Commutative Property of Multiplication
a⋅a⋅a=a3
Associative Property of Multiplication
The scale factor of two similar figures can be used to find the volume of one of the figures when the volume of the other figure is known.
The suitcase company Case-O-La produces snazzy suitcases of various sizes. When a large-sized suitcase is bought, the company offers its cabin-sized version at a discounted price to the same customer.
The large-sized suitcase has a height of 27 inches and a volume of 90 liters. If the cabin-sized suitcase has a height of 18 inches, determine its volume. Round the answer to one decimal place.If the scale factor of two similar figures is ba, then the ratio of their volumes is (ba)3.
The suitcases can be considered as two similar rectangular prisms with heights 27 and 18 inches.
Similar solids have the same shape and all of their corresponding sides are proportional. The ratio of the corresponding linear dimensions of the similar solids is the scale factor.ba=b/9a/9
LHS3=RHS3
(ba)m=bmam
Calculate power
ba=a:b
LHS⋅90=RHS⋅90