Perimeter, area and volume: changes in scale

Key Notes :

🔹Meaning of Scale

Scale means enlarging (making bigger) or reducing (making smaller) a shape while keeping the same proportions.

When dimensions (length, width, height) of a figure are multiplied by a scale factor, its perimeter, area, and volume also change.

🔹Effect of Scale Factor on Perimeter, Area, and Volume

👉 Suppose the scale factor = k
(Every length is multiplied by k)

Perimeter (1-D Measurement)

  • Perimeter is directly proportional to the scale factor.
  • New Perimeter = k × Original Perimeter

Area (2-D Measurement)

  • Area is proportional to the square of the scale factor.
  • New Area = k² × Original Area

Volume (3-D Measurement)

  • Volume is proportional to the cube of the scale factor.
  • New Volume = k³ × Original Volume
🔹 Examples

Example 1: Perimeter

  • A square has side = 5 cm.
  • Scale factor = 3.
  • New side = 5 × 3 = 15 cm.
  • Original Perimeter = 4 × 5 = 20 cm.
  • New Perimeter = 3 × 20 = 60 cm. ✅

Example 2: Area

  • Rectangle = 6 cm × 4 cm.
  • Scale factor = 2.
  • New dimensions = 12 cm × 8 cm.
  • Original Area = 6 × 4 = 24 cm².
  • New Area = 2² × 24 = 4 × 24 = 96 cm². ✅

Example 3: Volume

  • Cube with side = 2 cm.
  • Scale factor = 3.
  • New side = 6 cm.
  • Original Volume = 2³ = 8 cm³.
  • New Volume = 3³ × 8 = 27 × 8 = 216 cm³. ✅
🔹Key Points to Remember
  • Perimeter → multiplies by k
  • Area → multiplies by k²
  • Volume → multiplies by k³
  • As shapes grow larger, area and volume grow much faster than perimeter.
🔹 Real-Life Applications
  • Maps (scale drawings).
  • Models of buildings, cars, and machines.
  • Enlarging/reducing images in printing or designing.
✅ Summary:

When a figure is enlarged or reduced by a scale factor (k):

  • Perimeter × k
  • Area × k²
  • Volume × k³

Learn with an example

▶️ Look at this rectangular prism:

If the width is doubled, then which of the following statements about its volume will be true?

Look at this rectangular prism:

  • The new volume will be 3 times the old volume.
  • The new volume will be 2 times the old volume.
  • The new volume will be 12 of the old volume.
  • The new volume will be 4 times the old volume.

You can solve this problem without using the measurements given in the diagram.

The original rectangular prism had this volume:

V = lwh

The new rectangular prism will have 2 times the width. Since the original width was w, the new width will be 2w. Calculate the volume:

V = l(2w)h

= 2lwh

Divide the new volume by the original volume and simplify.

new volume/ original volume = 2lwh / lwh

= 2

The new volume will be 2 times the old volume.

▶️ Look at this cube:

If the side lengths are tripled, then which of the following statements about its surface area will be true?

  • The new surface area will be 3 times the old surface area
  • The new surface area will be 1/4 of the old surface area
  • The new surface area will be 27 times the old surface area.
  • The new surface area will be 9 times the old surface area

You can solve this problem without using the measurements given in the diagram.

The original cube had this surface area:

S = 6s2

The new cube will have sides that are 3 times as long. Since the original side lengths were s, the new side lengths will be 3s. Calculate the surface area:

S = 6(3s)2

= 6 · 9s2

= 54s2

Divide the new surface area by the original surface area and simplify.

new surface area / original surface area = 54s2 / 6s2

= 9

The new surface area will be 9 times the old surface area.

▶️ Look at this square:

If the side lengths are doubled, then which of the following statements about its perimeter will be true?

  • The new perimeter will be 2 times the old perimeter.
  • The new perimeter will be 3 times the old perimeter.
  • The new perimeter will be 1/2 of the old perimeter.
  • The new perimeter will be 4 times the old perimeter.

You can solve this problem without using the measurements given in the diagram.

The original square had this perimeter:

P = 4s

The new square will have sides that are 2 times as long. Since the original side lengths were s, the new side lengths will be 2s. Calculate the perimeter:

P = 4(2s)

= 8s

Divide the new perimeter by the original perimeter and simplify.

new perimeter / original perimeter = 8s/4s

= 2

The new perimeter will be 2 times the old perimeter.

Let’s practice!🖊️