11.10 Similarity in solid figures
Volume of similar 3D figures
Suppose we want to determine how the volume scale factor of a three-dimensional figure might change depending on the length scale factor $k$k.
Exploration
Given a cube of side length $a$a units, its volume would be $a^3$a3 units3.
Again, let's say we want to scale it by a length scale factor of $k>0$k>0.
The side length of our new cube is $a\times k$a×k units. We can again use this to figure out the new volume.
| New Area | $=$= | $\left(a\times k\right)\times\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k)×(a×k) | units3 |
| $=$= | $\left(a\times k\right)^3$(a×k)3 | units3 | |
| $=$= | $a^3\times k^3$a3×k3 | units3 |
Hence, we know that if we scale any cube by length scale factor $k$k, the volume scale factor will always be $k^3$k3.
For any figure that is scaled by a length scale factor of $k$k, the volume will scale by a volume scale factor of $k^3$k3.
Practice questions
Question 1
Consider the two similar trapezoidal prisms.
Find the length scale factor, going from the smaller prism to the larger prism.
Find the volume scale factor, going from the smaller prism to the larger prism.
Question 2
Consider the two similar spheres shown. The smaller sphere has radius $3$3 cm while the larger sphere has a radius of $12$12 cm.
Find the volume of Sphere A, in simplest exact form.
Find the volume of Sphere B, in simplest exact form.
What is the ratio of the volume of Sphere A to Sphere B?
Is it true that if the matching dimensions of two similar figures are in the ratio $m$m:$4$4, then their volumes are in the ratio $m^3$m3:$4^3$43
True
AFalse
B
Surface area of similar 3D figures
Given what we've previously discovered about the linear scale factor, we can find the relationship between the linear scale factor and surface area scale factor of a solid.
Given what we know about areas of similar figures, similar 2D shapes with a length scale factor $k$k will have an area scale factor $k^2$k2,
Given that the surface area of a shape is just all the area sum of all the little squares that fit on its surface, whenever we scale a solid using a linear scale factor, all the little square side lengths will scale also.
Hence, the total surface area sum will just scale by $k^2$k2, as was the case in two dimensions!
For any three dimensional figure that is scaled by a linear scale factor of $k$k, the surface area will scale by a surface area scale factor of $k^2$k2.
Question 3
Consider the rectangular prisms A and B shown.
Find the surface area of Rectangular Prism A.
Find the surface area of Rectangular Prism B.
What is the ratio of the surface area of Rectangular Prism A to Rectangular Prism B?
Is it true that if the matching sides of two similar figures are in the ratio $m$m:$3$3, then their surface areas are in the ratio $m^2$m2:$3^2$32?
True
AFalse
B