Effects of Scaling on Area and Volume

SAT Geometry and Trigonometry

In this lesson:

• How area scales when dimensions are multiplied by
• How volume scales when dimensions are multiplied by
• Reverse scaling and partial scaling on the SAT

Your Learning Goals for This Lesson

By the end of this lesson, you will be able to:

1. Apply the rule for area scaling
2. Apply the rule for volume scaling
3. Solve SAT-style scaling problems
4. Work backward from a ratio to find the scale factor

What Happens When You Double a Side?

A rectangle has area .

Double both sides → rectangle: .

The area is 4 times larger — not 2 times larger.

Area multiplied by , not just .

The Rule: Area Scales by

When all linear dimensions are multiplied by , area multiplies by .

Applying the Rule: Circles

A circle has radius 5. Area .

The radius is tripled (). New radius .

Area is multiplied by — not by .

Quick Check: Area After Halving

A square has side length 8. Its area is 64.

The side is halved to 4 ().

What is the new area? Express as a fraction of the original.

Think before advancing…

The Rule: Volume Scales by

When all linear dimensions are multiplied by , volume multiplies by .

Why? Volume involves three dimensions. Each multiplies by , so:

Applying the Rule: Cylinder Doubled

A cylinder has , . .

All dimensions doubled (): new , new .

Volume multiplied by .

Applying the Rule: Sphere Tripled

A sphere has radius . Volume .

The radius is tripled (). New radius .

Volume multiplied by .

Quick Check: Sphere Volume Doubles When

The volume of a sphere is multiplied by 8. By what factor was the radius multiplied?

Think before advancing…

Reverse Scaling: Find the Scale Factor

• Area ratio given →
• Volume ratio given →

Example: Volumes in ratio .

Partial Scaling: Only One Dimension Changes

and rules require ALL dimensions to scale by .

Example: Cylinder radius doubled, height unchanged.

Volume ×4, not ×8 — height stayed the same.

Your Turn: Three Scaling Problems

Try these on your own:

1. A square's side triples. Area multiplied by what factor?
2. A cube's side doubles. Volume multiplied by what factor?
3. Two similar spheres have radii in ratio . Their volumes are in what ratio?

Pause and solve each before advancing.

Check Your Work: Scaling Practice Solutions

1. : area

2. : volume

3. :

Volumes in ratio .

Key Takeaways: Area and Volume Scaling

• ✓ Area scales by (2D)
• ✓ Volume scales by (3D)
• ✓ Reverse: area ratio → ; volume ratio →

Doubling a side → area ×4, not ×2

only when ALL dimensions scale by

Decision Guide: Full vs Partial Scaling

Ask first: Do ALL dimensions scale by ?

What You Will Learn Next

This topic connects to similar figures and unit conversions.

Key ideas ahead:

• Similar triangles and corresponding side ratios
• Applying scaling reasoning to real-world measurement
• SAT problems that combine multiple geometry concepts

These skills build directly on today's and rules.