Scale Factor

Area of $K L M N$	Area of $P Q R S$
$A_{1} = K L \cdot L M$	$A_{2} = P Q \cdot Q R$

Area of

K L M N

Area of

P Q R S

A_{1} = K L \cdot L M

A_{2} = P Q \cdot Q R

The statement will be proven for similar rectangular prisms, but this proof can be adapted to prove other similar solids. 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 volume of a rectangular prism is the product of its base area and its height.

Volume of Solid $A$	Volume of Solid $B$
$V_{1} = a_{1} \cdot a_{2} \cdot a_{3}$	$V_{2} = b_{1} \cdot b_{2} \cdot b_{3}$

By the definition of similar solids, the side lengths are proportional and equal to the scale factor

\frac{a}{b} .

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ \frac{a _{1}}{b _{1}} = \frac{a}{b} \frac{a _{2}}{b _{2}} = \frac{a}{b} \frac{a _{3}}{b _{3}} = \frac{a}{b} \Leftrightarrow ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ a_{1} = b_{1} \cdot \frac{a}{b} a_{2} = b_{2} \cdot \frac{a}{b} a_{3} = b_{3} \cdot \frac{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 .

V_{1} = a_{1} \cdot a_{2} \cdot a_{3}

SubstituteExpressions

Substitute expressions

V_{1} = (b_{1} \cdot \frac{a}{b}) (b_{2} \cdot \frac{a}{b}) (b_{3} \cdot \frac{a}{b})

▼

Simplify right-hand side

RemovePar

Remove parentheses

V_{1} = b_{1} \cdot \frac{a}{b} \cdot b_{2} \cdot \frac{a}{b} \cdot b_{3} \cdot \frac{a}{b}

CommutativePropMult

Commutative Property of Multiplication

V_{1} = \frac{a}{b} \cdot \frac{a}{b} \cdot \frac{a}{b} \cdot b_{1} \cdot b_{2} \cdot b_{3}

ProdToPowThreeFac

$a \cdot a \cdot a = a^{3}$

V_{1} = (\frac{a}{b})^{3} \cdot b_{1} \cdot b_{2} \cdot b_{3}

AssociativePropMult

Associative Property of Multiplication

V_{1} = (\frac{a}{b})^{3} (b_{1} \cdot b_{2} \cdot b_{3})

Notice that the expression on the right-hand side is

(\frac{a}{b})^{3}

times the volume of Solid

B .

V_{1} = (\frac{a}{b})^{3} (b_{1} \cdot b_{2} \cdot b_{3})

Substitute

$b_{1} \cdot b_{2} \cdot b_{3} = V_{2}$

V_{1} = (\frac{a}{b})^{3} V_{2}

DivEqn

$LHS / V_{2} = RHS / V_{2}$

\frac{V _{1}}{V _{2}} = (\frac{a}{b})^{3}

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.

\underline{Scale Factor} \frac{a}{b} \Rightarrow \underline{Volume Scale Factor} \frac{V _{1}}{V _{2}} = (\frac{a}{b})^{3}

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Scale Factors

Length Scale Factor

Similar Figures

Area Scale Factor

Example Considering Similar Figures

Areas of Similar Figures

Proof

Volume Scale Factor

Example Considering Similar Figures

Volumes of Similar Solids

Proof

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