Scaling Proportionality

This pattern tests how area, perimeter, and volume change when dimensions are scaled. The core rules are: if linear dimensions scale by a factor of $k$, then perimeter scales by $k$, area scales by $k^2$, and volume scales by $k^3$.

 

Perimeter Scales Linearly

If each side of a figure is multiplied by $k$, the perimeter is also multiplied by $k$.

Pentagon $ABCDE$ is similar to pentagon $FGHIJ$. Side $AB = 6$ and side $FG = 18$. If the perimeter of $ABCDE$ is 45, what is the perimeter of $FGHIJ$?

A) $15$
B) $57$
C) $135$
D) $405$

Scale factor $k = \dfrac{18}{6} = 3$. New perimeter $= 3 \times 45 = 135$. The answer is C. Option D multiplies by $k^2 = 9$ instead of $k$.

Quadrilateral $ABCD$ is similar to quadrilateral $EFGH$. Side $AB = 7$ and $EF = 28$. If the perimeter of $ABCD$ is 25, what is the perimeter of $EFGH$?

A) $60$
B) $100$
C) $175$
D) $400$

$k = \dfrac{28}{7} = 4$. Perimeter $= 4 \times 25 = 100$. The answer is B.

 

Finding a Corresponding Side from Similar Figures

Use the scale factor to find missing side lengths.

Triangle $FGH$ is similar to triangle $JKL$. $FG = 21$, $JK = 7$, and $JL = 9$. What is $FH$?

A) $9$
B) $14$
C) $27$
D) $63$

$k = \dfrac{FG}{JK} = \dfrac{21}{7} = 3$. Since $FH$ corresponds to $JL$: $FH = 3 \times 9 = 27$. The answer is C.

Equilateral triangle A has side length 8 inches. The perimeter of equilateral triangle B is 3 times the perimeter of A. What is the side length of triangle B?

A) $11$
B) $16$
C) $18$
D) $24$

If the perimeter is 3 times as large, each side is also 3 times as large: $3 \times 8 = 24$. The answer is D.

 

Area Scales by $k^2$

When linear dimensions scale by $k$, area scales by $k^2$. This is the most heavily tested rule.

Circle P has a radius 18 times the radius of circle Q. The area of circle P is $k$ times the area of circle Q. What is $k$?

A) $18$
B) $36$
C) $324$
D) $9$

Since radius scales by 18, area scales by $18^2 = 324$. The answer is C.

Two equilateral triangles are similar. The side of triangle X is 180 times the side of triangle Y. The area of X is $k$ times the area of Y. What is $k$?

A) $90$
B) $180$
C) $360$
D) $32{,}400$

$k = 180^2 = 32{,}400$. The answer is D. The shape doesn't matter — the area ratio always equals the square of the linear ratio.

Triangles PQR and STU are similar. Each side of STU is 4 times the corresponding side of PQR. The area of PQR is 18 square centimeters. What is the area of STU?

A) $16$
B) $72$
C) $288$
D) $4.5$

Area factor $= 4^2 = 16$. Area of STU $= 16 \times 18 = 288$. The answer is C.

 

Working Backward: Area Ratio to Linear Ratio

If you know the area ratio, take the square root to find the linear scale factor.

Two triangular flags are similar. Flag A has area 320 in$^2$ and Flag B has area 20 in$^2$. The longest side of Flag A is 48 inches. What is the longest side of Flag B?

A) $3$
B) $12$
C) $16$
D) $48$

Area ratio $= \dfrac{320}{20} = 16$, so linear ratio $= \sqrt{16} = 4$. Longest side of B $= \dfrac{48}{4} = 12$. The answer is B.

Trapezoid $EFGH$ is similar to trapezoid $JKLM$. Each side of $EFGH$ is $\dfrac{1}{3}$ the corresponding side of $JKLM$. If the area of $JKLM$ is 72 m$^2$, what is the area of $EFGH$?

A) $8$
B) $24$
C) $63$
D) $216$

Area factor $= \left(\dfrac{1}{3}\right)^2 = \dfrac{1}{9}$. Area $= \dfrac{72}{9} = 8$. The answer is A.

 

Scaling a Single Dimension (Area)

If each dimension is scaled by a fraction or percentage, the area scales by that factor squared.

A poster has area 900 in$^2$. A thumbnail is made with each dimension $\dfrac{1}{6}$ of the poster. What is the thumbnail area?

A) $5$
B) $25$
C) $75$
D) $150$

$\left(\dfrac{1}{6}\right)^2 = \dfrac{1}{36}$. Area $= \dfrac{900}{36} = 25$. The answer is B.

A square patio has area 240 ft$^2$. Each side is decreased by 30%. What is the new area?

New side $= 0.70 \times$ old side. New area $= (0.70)^2 \times 240 = 0.49 \times 240 = 117.6$ ft$^2$.

A circular flower bed has area 250 m$^2$. The radius is increased by 30%. What is the new area?

New radius $= 1.30 \times$ old radius. New area $= (1.30)^2 \times 250 = 1.69 \times 250 = 422.5$ m$^2$.

 

Volume Scales by $k^3$

When linear dimensions scale by $k$, volume scales by $k^3$.

A spherical balloon has volume 1,200 cm$^3$. Its radius decreases by 10%. What is the new volume?

New radius $= 0.90 \times$ old radius. New volume $= (0.90)^3 \times 1{,}200 = 0.729 \times 1{,}200 = 874.8$ cm$^3$.

 

Surface Area Ratio to Volume Ratio

If you know the surface area ratio, take the square root for the linear ratio, then cube it for the volume ratio.

Pyramid P is similar to pyramid Q. Surface areas are 40 in$^2$ and 360 in$^2$. Volume of Q is 810 in$^3$. What is the sum of the volumes?

Surface area ratio $= \dfrac{360}{40} = 9$. Linear ratio $= \sqrt{9} = 3$. Volume ratio $= 3^3 = 27$. Volume of P $= \dfrac{810}{27} = 30$. Sum $= 810 + 30 = 840$.

Summary of scaling rules:

Linear dimensions $\times k$ $\Rightarrow$ Perimeter $\times k$, Area $\times k^2$, Volume $\times k^3$.

 

What to Do on Test Day

• The three scaling rules — memorize these:
• Linear dimensions $\times k$ → Perimeter $\times k$
• Linear dimensions $\times k$ → Area $\times k^2$
• Linear dimensions $\times k$ → Volume $\times k^3$
• Working backward:
• If area ratio is given, take $\sqrt{}$ to get the linear ratio.
• If volume ratio is given, take $\sqrt[3]{}$ to get the linear ratio.
• If surface area ratio is given, take $\sqrt{}$ for the linear ratio, then cube for volume.
• Percentage changes: "Increased by 30%" means the new dimension is $1.30$ times the old. "Decreased by 20%" means $0.80$ times. Then square (for area) or cube (for volume).
• The shape doesn't matter. The scaling rules apply to all similar figures — circles, triangles, pentagons, spheres, anything. Don't get distracted by the shape.
• Biggest trap: Using $k$ when you need $k^2$ (or $k^3$). If the side ratio is 3, the area ratio is 9, not 3.