Surface Area Ratios of Similar Solids
Surface Area Ratios of Similar Solids is a Grade 8 math skill from Big Ideas Math, Course 3, Chapter 8: Volume and Similar Solids. If two similar solids have a scale factor of a:b, their surface areas are in the ratio a^2:b^2, because surface area is a two-dimensional measure that scales by the square of the linear scale factor. Students apply this property to find unknown surface areas of similar three-dimensional figures such as pyramids and cones.
Key Concepts
If two similar solids have a scale factor of $a:b$, then the ratio of their corresponding surface areas is $a^2:b^2$.
$$\frac{\text{Surface Area of Solid A}}{\text{Surface Area of Solid B}} = \left(\frac{\text{Corresponding length of A}}{\text{Corresponding length of B}}\right)^2 = \frac{a^2}{b^2}$$.
Common Questions
What is the ratio of surface areas for similar solids?
If the scale factor between two similar solids is a:b, then their surface areas are in the ratio a^2:b^2, because surface area scales by the square of the linear scale factor.
How do you find the surface area of a similar solid given the scale factor?
Set up a proportion using the squared scale factor: SA2/SA1 = (b/a)^2, then solve for the unknown surface area.
Why is the surface area ratio the square of the scale factor?
Surface area is measured in square units, so when all linear dimensions are multiplied by a factor, each face area is multiplied by the square of that factor.
Where are similar solids and surface area ratios in the Grade 8 curriculum?
Big Ideas Math, Course 3, Chapter 8: Volume and Similar Solids covers surface area ratios of similar solids in the Grade 8 math curriculum.