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← Area and Volume Scale Factors

Reference

Scale Factors

Concept

Length Scale Factor

The length scale factor of a scale drawing or scale model is the ratio of a length on the drawing or model to the corresponding actual length where both lengths have the same units of measure.

Length scale factor = Length on model/Actual length

Since it is a ratio, the length scale factor can also be written using colon notation. However, it is usually written as a constant that describes the relationship between the dimensions of the scale drawing or scale model and the actual dimensions.

Similar Figures

The length scale factor definition also applies for similar figures. Examine these two similar triangles.

Triangle ABC, with side length AC = 2.5, is similar to triangle PQR, which has a side length PR = 5.

The length scale factor used to get △ PQR from △ ABC is the ratio between a measure on the new figure, △ PQR, and the corresponding measure on the initial figure, △ ABC. Here, PR and AC are corresponding sides. Therefore, the length scale factor is the ratio between PR and AC. Length scale factor &= PR/AC = 5/2.5 &⇓ Length scale factor &= 2

Every side length in △ PQR is twice the corresponding side length in △ ABC. Notice that the length scale factor for getting △ ABC from △ PQR is 12.

Concept

Area Scale Factor

For similar figures, the ratio between their areas is called the area scale factor.

Area Scale Factor Area of Figure2/Area of Figure1 or Area of Figure2: Area of Figure1

The scale factor of the areas of similar figures can also be calculated by squaring the length scale factor of the figures.

Example Considering Similar Figures

Examine these two similar triangles.

Triangle ABC with area of 3 units square and a triangle PQR with area of 27 units squared, which are similar.

It can be seen that the areas of △ ABC and △ PQR are 3 and 27 square units, respectively. This is enough information to calculate the area scale factor.

Area Scale Factor=Area of Figure2/Area of Figure1

Area of Figure1= 3, Area of Figure2= 27

Area Scale Factor=27/3

Calculate quotient

Area Scale Factor=9

Rule

Areas of Similar Figures

If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Similar Quadrilaterals

Let KLMN and PQRS be similar figures, and A_1 and A_2 be their respective areas. The length scale factor between corresponding side lengths is ab. Here, the following conditional statement holds true.

KLMN ~ PQRS ⇒ A_1/A_2 = (a/b )^2

Proof

The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.

The area of a rectangle is the product of its length and its width.

Area of KLMN	Area of PQRS
A_1 = KL* LM	A_2 = PQ * QR

By the definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor ab. KL/PQ= a/b [1.1em] LM/QR=a/b ⇔ KL = PQ * a/b [1.1em] LM = QR * a/b The next step is to substitute the expressions for KL and LM into the formula for A_1, which represents the area of KLMN.

A_1 = KL* LM

KL= PQ * a/b, LM= QR * a/b

A_1 = ( PQ * a/b) ( QR * a/b)

▼

Simplify right-hand side

Remove parentheses

A_1 = PQ * a/b * QR * a/b

CommutativePropMult

Commutative Property of Multiplication

A_1 = a/b * a/b * PQ * QR

ProdToPowTwoFac

a* a=a^2

A_1 = (a/b )^2 * PQ * QR

AssociativePropMult

Associative Property of Multiplication

A_1 = (a/b )^2(PQ * QR)

Notice that the expression on the right-hand side is ( ab )^2 times the area of PQRS, or A_2.

A_1 = (a/b )^2( PQ* QR)

PQ* QR= A_2

A_1 = (a/b )^2 A_2

.LHS /A_2.=.RHS /A_2.

A_1/A_2 = (a/b )^2

This proof has shown that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths. This ratio is also called the area scale factor.

Scale Factor & & Area Scale Factor a/b & ⇒ & A_1/A_2 = (a/b )^2

Concept

Volume Scale Factor

For similar solids, the ratio between their volumes is called the volume scale factor.

Volume Scale Factor Volume of Solid2/Volume of Solid1 or Volume of Solid2 : Volume of Solid1

The scale factor of the volumes of similar solids can also be calculated by cubing the length scale factor of the solids.

Example Considering Similar Figures

Examine these two similar cuboids.

similar prism

The volume scale factor can be calculated using the volumes of these two similar figures.

Volume Scale Factor=Volume of Solid2/Volume of Solid1

Volume of Solid1= 10, Volume of Solid2= 80

Volume Scale Factor=80/10

Calculate quotient

Volume Scale Factor=8

Rule

Volumes of Similar Solids

If two figures are similar, then the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths.

Two similar solids

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

Proof

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* 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.

V_1 = a_1 * a_2 * a_3

SubstituteExpressions

Substitute expressions

V_1 = ( b_1 * a/b) ( b_2 * a/b) ( b_3 * a/b)

▼

Simplify right-hand side

Remove parentheses

V_1 = b_1 * a/b * b_2 * a/b * b_3 * a/b

CommutativePropMult

Commutative Property of Multiplication

V_1 = a/b * a/b * a/b * b_1 * b_2 * b_3

ProdToPowThreeFac

a* a* a=a^3

V_1 = (a/b )^3 * b_1 * b_2 * b_3

AssociativePropMult

Associative Property of Multiplication

V_1 = (a/b )^3 (b_1 * b_2 * b_3)

Notice that the expression on the right-hand side is ( ab )^3 times the volume of Solid B.

V_1 = (a/b )^3 (b_1 * b_2 * b_3)

b_1 * b_2 * b_3= V_2

V_1 = (a/b )^3 V_2

.LHS /V_2.=.RHS /V_2.

V_1/V_2 = (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.

Scale Factor & & Volume Scale Factor a/b & ⇒ & V_1/V_2 = (a/b )^3

Exercises

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