Question

Create and provide the dimensions for two similar figures of your choosing.
Part 2: What is the similarity ratio of these figures along with the ratio of their surface area and volume?
Part 3: Show your work, either using the actual volumes or using the formula, that the volume ratio is true.

Answer 1

Let's use two spheres, sphere 1 and sphere 2 where the radius \[r _{1}\]of sphere 1 is 1 cm and the radius\[r _{2}\] of sphere 2 is 2 cm. The ratio of the radii of the two spheres is\[r _{2}/r _{1} =2/1=2\] The surface area of a sphere is given by\[A= 4 \times pi \times r ^{2}\]

Answer 2

The surface area of sphere 1 would be\[A _{1}=4 \times pi \times( r _{1})^2 =4 \times pi \times 1^2 =4 pi\] and likewise the surface area of sphere 2 would be \[A _{2}=4 \times pi \times( r _{2})^2 =4 \times pi \times 2^2 =16 pi\] so the ratio of surface areas would be\[A _{2}/A _{1}=16 pi/4 pi = 4\]

Answer 3

The volume V of a sphere is given by\[V = (4/3) \times pi \times r^3\] so for sphere 1
\[V _{1}=(4/3) \times pi \times (r _{1})^3 = (4/3) \times pi \times (1)^3= (4/3) pi\]and for sphere 2\[V _{2}=(4/3) \times pi \times (r _{2})^3 = (4/3) \times pi \times (2)^3= (4/3) \times pi \times 8\] and the ratios of the two volumes \[V _{2}/V _{1}= (4/3pi \times 8)/(4/3pi) =8\]would be