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Surface Areas of Similar Solids
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Rule
Surface Areas of Similar Solids
If two figures are similar, then the ratio of their surface areas is equal to the square of the ratio of their corresponding side lengths.
Let Solid A and Solid B be similar solids and SA_1 and SA_2 be their respective surface areas. The length scale factor between corresponding linear measures is ab. Given these characteristics, the following conditional statement holds true.
SolidA ~ SolidB ⇒ SA_1/SA_2 = (a/b)^2
Proof
The statement will be proven for similar rectangular prisms, but this proof can be adapted to prove other similar solids as well. As shown in the diagram, let a_1, a_2, and a_3 be the dimensions of Solid A and b_1, b_2, and b_3 be the dimensions of Solid B.
The surface area of a rectangular prism is the sum of the lateral area and the combined areas of the two identical bases. The lateral area of a rectangular prism consists of its four rectangular side areas. Notice that the areas of opposite faces are congruent.
| Surface Area of Solid A | Surface Area of Solid B |
|---|---|
| SA_1 =2( a_1* a_2 + a_1 * a_3+ a_2 * a_3 ) | SA_2 = 2( b_1* b_2 + b_1 * b_3+ 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 SA_1, the surface area of Solid A.
SA_1 = 2(a_1 * a_2 + a_1 * a_3+a_2 * a_3 )
SubstituteExpressions
Substitute expressions
SA_1 = 2 (( b_1 * a/b * b_2 * a/b ) + ( b_1 * a/b * b_3 * a/b) + ( b_2 * a/b * b_3 * a/b ) )
▼
Simplify right-hand side
CommutativePropMult
Commutative Property of Multiplication
SA_1 = 2 ((a/b * a/b * b_1 * b_2 ) + (a/b * a/b * b_1 * b_3 ) + ( a/b * a/b * b_2 * b_3 ) )
RemovePar
Remove parentheses
SA_1 = 2 (a/b * a/b * b_1 * b_2 + a/b * a/b * b_1 * b_3 + a/b * a/b * b_2 * b_3 )
ProdToPowTwoFac
a* a=a^2
SA_1 = 2 ( (a/b )^2 * b_1 * b_2 + (a/b )^2 * b_1 * b_3 + (a/b )^2 * b_2 * b_3 )
FactorOut
Factor out (a/b )^2
SA_1 = (a/b )^2 * 2 ( b_1 * b_2 + b_1 * b_3 + b_2 * b_3 )
Notice that the expression on the right-hand side is ( ab )^2 times the surface area of Solid B.
SA_1 = (a/b )^2 * 2 ( b_1 * b_2 + b_1 * b_3 + b_2 * b_3 )
Substitute
2 ( b_1 * b_2 + b_1 * b_3 + b_2 * b_3 )= SA_2
SA_1 = (a/b )^2 * SA_2
DivEqn
.LHS /SA_2.=.RHS /SA_2.
SA_1/SA_2 = (a/b )^2
As shown, the ratio of the surface areas of the similar prisms is equal to the square of the ratio of their corresponding linear measures.
Scale Factor & & Surface Area Scale Factor a/b & ⇒ & SA_1/SA_2 = (a/b )^2
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