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 and the volume of the smaller pyramid is cubic centimeters, what is the volume of the larger model?
The theorem will be proven for similar rectangles. The proof can be adapted to other similar figures.
The area of a rectangle is the product of its length and its width.
|Area of||Area of|
Dominika is making a flyer for a concert and wants to compare the areas of the two pieces of paper. The widths of and papers are and centimeters, respectively.
The two pieces of paper are similar and two corresponding sides measure centimeters and centimeters.
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 has an area of square inches, and Figure has an area of square inches.
Determine the linear scale factor of the shape on the right to the shape on the left.
Let Solid and Solid be similar solids, and and be their respective volumes. The length scale factor between corresponding linear measures is Given those characteristics, the following conditional statement holds true.
The theorem will be proven for similar rectangular prisms. Take into consideration that the proof can be adapted to prove other similar solids as well. As shown in the diagram, let and be the dimensions of Solid and and be the dimensions of Solid
The volume of a rectangular prism is the product of its base area and its height.
|Volume of Solid||Volume of Solid|
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 suitcases can be considered as two similar rectangular prisms with heights and inches.
If the radius of the Earth is about kilometers, help Mark find the radius of the Sun.
Write in scientific notation
Write in scientific notation
The corresponding faces of two similar three-dimensional figures are also similar. Subsequently, the ratio of the areas of the corresponding faces is proportional to the square of the length scale factor of the figures.
Dylan has a golden retriever and a chihuahua. He buys two similar doghouses, whose corresponding side lengths are proportional. When he is about to finish painting the doghouses, he realizes that there is not enough paint for the front face of the small doghouse!
Dylan knows the volumes of each doghouse. They are about and cubic inches. He also knows that the front face of the big doghouse has an area of square inches. Help Dylan determine the area of the front face of the small doghouse. This will help him determine how much more paint to buy. Round the answer to the nearest integer.
Like similar solids, if the corresponding linear measures of two composite solids are proportional, the composite solids are said to be similar. Therefore, their length scale factor can be determined and used to find certain characteristics of the shapes.
The amount of material used to construct the larger silo is three times that of the smaller one. That is, the surface area of the larger silo is three times as large as the surface area of the smaller silo. These two silos can be considered to be similar solids. Furthermore, each silo is composed of a cone and a cylinder as shown.
The given silos are similar composite solids. Since they are similar, their corresponding linear measures are proportional. Therefore, each silo can be considered as a whole. To find the volume of the smaller silo, these steps will be followed.
It is given that the surface area of the larger silo is three times as large as the surface area of the smaller silo.
In this course, the relationships between the length scale factor, area scale factor and volume scale factor have been discussed. If the scale factor between two similar figures is then the ratio for their areas and volumes can be expressed as the table shows.
|Length Scale Factor||Area Scale Factor||Volume Scale Factor|
Considering these expressions, the challenge presented at the beginning can be solved with more confidence.
If the scale factor of two similar figures is then the ratio of their volumes is