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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 V_1 and V_2 be their respective volumes. The length scale factor between corresponding linear measures is ab. Given these characteristics, the following conditional statement holds true.
SolidA ~ SolidB ⇒ V_1/V_2 = (a/b)^3
The volume of a rectangular prism is the product of its base area and its height.
| Volume of Solid A | Volume of Solid B |
|---|---|
| V_1 = a_1* a_2 * a_3 | V_2 = b_1 * b_2 * b_3 |
By the definition of similar solids, the side lengths are proportional and equal to the scale factor ab. a_1/b_1=a/b [1.1em] a_2/b_2=a/b [1.1em] a_3/b_3=a/b ⇔ a_1 = b_1 * a/b [1.1em] a_2 = b_2 * a/b [1.1em] a_3 = b_3 * a/b The next step is to substitute the expressions for a_1, a_2, and a_3 into the formula for V_1, the volume of Solid A.
Substitute expressions
Remove parentheses
Commutative Property of Multiplication
a* a* a=a^3
Associative Property of Multiplication
Notice that the expression on the right-hand side is ( ab )^3 times the volume of Solid B.
b_1 * b_2 * b_3= V_2
.LHS /V_2.=.RHS /V_2.
As shown, the ratio of the volumes of the similar prisms is equal to the cube of the ratio of their corresponding linear measures. This ratio is also called the volume scale factor.
Scale Factor & & Volume Scale Factor a/b & ⇒ & V_1/V_2 = (a/b )^3