G.11(D)·Geometry·Similarity & MeasurementReadiness

Scale Factors for Perimeter, Area, and Volume

Relate linear scale factors to changes in perimeter, area, and volume.

Students describe how scaling dimensions changes perimeter, area, and volume, using k, k², and k³ for similar figures and multiplying only the factors of dimensions that actually change.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice45 minutes

Learn the concept

Scale factor patterns

If every linear dimension of a figure is multiplied by k, perimeter is multiplied by k, area by $k^2$ , and volume by $k^3$ . A scale factor less than 1 shrinks every measurement.

Why area squares

Area multiplies two lengths. If both are multiplied by k, the area is multiplied by $k\cdot k=k^2$ . A square with one-third the side length has $\left(\frac{1}{3}\right)^2=\frac{1}{9}$ of the area.

One dimension or all

The $k^2$ rule applies only to similar figures, where every dimension is scaled. If only one dimension changes, area changes by that one factor. Given an area ratio, take its square root to find the linear scale factor.

Perimeter scales by k

Perimeter adds side lengths. Multiplying every side by k gives $ka+kb+kc=k(a+b+c)$ , so the new perimeter is k times the original perimeter, not $k^2$ times it. This works for any polygon whose side lengths all scale by the same factor.

Scale only the base dimensions

For a rectangular prism, $V=lwh$ . If length and width each double but height stays fixed, the volume factor is $2\cdot2\cdot1=4$ . More generally, multiply the individual dimension factors. A pyramid also quadruples in volume when both base dimensions double and its perpendicular height stays fixed, because $V=\frac{1}{3}Bh$ and B quadruples.

Similar solids scale in three dimensions

For similar solids, every corresponding linear dimension scales by k, so volume scales by $k^3$ . Doubling every dimension multiplies volume by 8. Doubling only two base dimensions multiplies volume by 4 and does not produce a similar solid when height stays fixed.

Worked example

A circle's radius changes from 4 cm to 10 cm. By what factor does its area change?

The scale factor is 10/4 = 2.5.
The area changes by $2.5^2=6.25$ .
Check: $\pi(4)^2=16\pi$ and $\pi(10)^2=100\pi$ ; $\frac{100\pi}{16\pi}=6.25$ .

The area is multiplied by 6.25.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

8th Grade STAAR Practice Dilations and Measurements (8.10D - #5)

Five Minute Math · Aaron Daffern, Dallas ISD

Model how dilations change linear and area measurements of two-dimensional figures.

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Texas public school video

Geometry: Comparing Volumes of Similar Figures

UT IPSI · Kid2Kid / Texas Education Agency

Compare measurements of similar figures to see how scale factors affect size.

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Try it yourself

TEKSForge practice · unscored

1. The side length of a square doubles. How does its area change?
Show answer and reasoning
It is multiplied by 4.
The linear scale factor is 2, so the area changes by $2^2=4$ .
2. Two similar rectangles have an area ratio of 9 : 25. What is the ratio of their perimeters?
Show answer and reasoning
3 : 5
Take the square root of the area ratio to find the linear scale factor: $\sqrt{9}:\sqrt{25}=3:5$ .
3. A triangle has perimeter 30 cm. Every side is multiplied by 3. Find the new perimeter.
Show answer and reasoning
90 cm
Perimeter scales by the linear factor: 30(3) = 90.
4. A rectangular prism has volume 120 cubic units. Find the new volume if both base dimensions double and height stays fixed. Then find the volume if all three dimensions double instead.
Show answer and reasoning
480 cubic units; 960 cubic units
Base-only scaling gives $120(2)(2)(1)=480$ . Scaling every dimension gives $120(2^3)=960$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 12 · 1 point

The side length of a smaller square is one-third the side length of a larger square. Which statement describes the area of the smaller square?

The area of the smaller square is $\frac{1}{27}$ the area of the larger square.
The area of the smaller square is $\frac{1}{6}$ the area of the larger square.
The area of the smaller square is $\frac{1}{9}$ the area of the larger square.
The area of the smaller square is $\frac{1}{3}$ the area of the larger square.

Show STAAR answer and explanation

The area of the smaller square is $\frac{1}{9}$ the area of the larger square.

Area scales by the square of the linear scale factor: $(1/3)^2=1/9$ . If the large side is s, the areas are $s^2$ and $(s/3)^2=s^2/9$ .

The area of the smaller square is $\frac{1}{27}$ the area of the larger square.: One twenty-seventh would be the volume ratio for similar cubes, not the area ratio for squares.

The area of the smaller square is $\frac{1}{6}$ the area of the larger square.: The scale factor must be squared, not doubled: $(1/3)^2=1/9$ .

The area of the smaller square is $\frac{1}{3}$ the area of the larger square.: One third is the side-length ratio; area requires squaring that ratio.

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Archived released STAAR Geometry EOC · 2013 · Item 20 · 1 point

The volume of a rectangular prism is 960 cubic inches. If the dimensions of the base are doubled and the height remains the same to create a new prism, what will be the volume of the new rectangular prism in cubic inches?

Show STAAR answer and explanation

3840

Both base dimensions double, so base area increases by $2\cdot2=4$ . With unchanged height, volume is $4(960)=3{,}840$ cubic inches.

1920: This doubles only one base dimension; both the length and width double.

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Archived released STAAR Geometry EOC · 2013 · Item 46 · 1 point

A triangle is enlarged by multiplying each of its dimensions by 4. Based on this information, which of the following statements is true?

The perimeter of the new triangle is 12 times the perimeter of the original triangle.
The perimeter of the new triangle is 16 times the perimeter of the original triangle.
The perimeter of the new triangle is 18 times the perimeter of the original triangle.
The perimeter of the new triangle is 4 times the perimeter of the original triangle.

Show STAAR answer and explanation

The perimeter of the new triangle is 4 times the perimeter of the original triangle.

If the original sides are a, b, and c, the new perimeter is $4a+4b+4c=4(a+b+c)$ . Perimeter scales by the linear factor 4.

The perimeter of the new triangle is 12 times the perimeter of the original triangle.: This multiplies the scale factor by the number of sides, counting the side sum twice.

The perimeter of the new triangle is 16 times the perimeter of the original triangle.: 16 is the area scale factor, not the perimeter scale factor.

The perimeter of the new triangle is 18 times the perimeter of the original triangle.: 18 is not the linear scale factor; summing scaled sides factors out exactly 4.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Explain why area changes by the square of a linear scale factor.
Apply k, k², and k³ to perimeter, area, and volume problems.
Distinguish scaling all dimensions from scaling only base dimensions while height stays fixed.

Instructional sequence

1
Engage
5 min
Build a 1-by-1 square and a 3-by-3 square with tiles. Ask how many times as many tiles the larger square needs.
2
Investigation
12 min
Scale rectangles and cubes by 2 and 3, recording the changes in perimeter, area, and volume.
3
Direct instruction
10 min
Generalize the k, k², and k³ patterns and compare scaling one dimension with scaling all dimensions.
4
Guided practice
12 min
Solve forward and reverse problems, including area ratios to linear scale factors.
5
Closure
6 min
Students complete the exit ticket using a sketch or tiles.

Misconceptions to address

““If the sides are one-third as long, the area is one-third as large.””

Teacher move: Area uses two dimensions, so multiply by the scale factor twice. Use tiles to show the squared change.

““Volume changes by the square of the scale factor.””

Teacher move: Volume has three dimensions, so it changes by k³.

““Doubling both base dimensions always makes volume eight times as large.””

Teacher move: List the factors for length, width, and height. With unchanged height, the product is 2 × 2 × 1 = 4; eight requires all three dimensions to double.

Materials

•Square tiles or grid paper
•Similar-figure cards
•Calculator

Teaching tips

Have students count tiles or unit cubes before stating a general rule.
Write the dimension count next to each measurement: perimeter 1D, area 2D, volume 3D.
For reverse problems, take the square root of an area ratio to find k.

Exit ticket · 3–5 min

A rectangle's length and width are each multiplied by $\frac{1}{2}$ . What fraction of the original area does the new rectangle have?

Show expected answer

$\left(\frac{1}{2}\right)^2=\frac{1}{4}$ , so it has one-fourth of the original area.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Scale factor patterns

Why area squares

One dimension or all

Perimeter scales by k

Scale only the base dimensions

Similar solids scale in three dimensions

Worked example

Watch the lesson

8th Grade STAAR Practice Dilations and Measurements (8.10D - #5)

Geometry: Comparing Volumes of Similar Figures

Try it yourself

1. The side length of a square doubles. How does its area change?

2. Two similar rectangles have an area ratio of 9 : 25. What is the ratio of their perimeters?

3. A triangle has perimeter 30 cm. Every side is multiplied by 3. Find the new perimeter.

4. A rectangular prism has volume 120 cubic units. Find the new volume if both base dimensions double and height stays fixed. Then find the volume if all three dimensions double instead.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 12 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 20 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 46 · 1 point

Learning objectives

Instructional sequence

Engage

Investigation

Direct instruction

Guided practice

Closure

Misconceptions to address