Virginia SOL Math / Geometry / Two- & Three-Dimensional Figures

G.DF.2

Changing a Figure's Dimensions

What you'll learn

  • Describe how scaling a length affects perimeter, area, and volume
  • Use the scale factor k for area (k^2) and volume (k^3)
  • Solve problems comparing similar figures and solids

Words to know

Scale factor —
the ratio k by which each length is multiplied
Similar solids —
solids with the same shape and proportional dimensions
Area ratio —
the square of the scale factor, k^2
Volume ratio —
the cube of the scale factor, k^3

Why it matters Understanding scaling explains why a doubled recipe needs four times the pan area, and why a slightly bigger box holds much more.

Worked examples

Model ItI do

The edge of a cube doubles from 2 to 4 (scale factor 2). How do the surface area and volume change?

Surface area scales by k^2 = 2^2 = 4, so it becomes 4 times as large. Volume scales by k^3 = 2^3 = 8, so it becomes 8 times as large.

Try It TogetherWe do

A rectangle's sides are tripled (scale factor 3). By what factor does its area grow?

Area scales by the square of the scale factor: 3^2 = 9. The area becomes 9 times as large.

Now YouYou do

Two similar solids have a scale factor of 2. If the small solid has volume 5, find the large solid's volume.

Multiply the volume by k^3. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.If the lengths of a figure double, its area is multiplied by what factor?

  2. 2.If the lengths of a solid double, its volume is multiplied by what factor?

  3. 3.A scale factor of 3 multiplies area by what factor?

On Level

The core Quick Check from the book.

  1. 1.If lengths double, area is multiplied by what factor?

  2. 2.If lengths double, volume is multiplied by what factor?

  3. 3.A scale factor of 3 multiplies area by _____.

  4. 4.A scale factor of 2 multiplies volume by _____.

  5. 5.Two similar solids have scale factor 2 and small volume 4. Find the large volume.

Challenge

Stretch your thinking.

  1. 1.Two similar solids have a scale factor of 3. The small solid has volume 4. Find the large solid's volume.

  2. 2.Two similar solids have surface areas 16 and 144. Find the scale factor between them.

  3. 3.Two similar solids have a scale factor of 2. The large solid has volume 40. Find the small solid's volume.