Square vs Cube Intuition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
A side length doubles from 3 to 6. By what factor does the area grow? By what factor does the volume grow?

Solution

  1. 1
    Original area: \(3^2 = 9\). New area: \(6^2 = 36\). Factor: \(36/9 = 4\).
  2. 2
    Original volume: \(3^3 = 27\). New volume: \(6^3 = 216\). Factor: \(216/27 = 8\).
  3. 3
    When length doubles (×2), area grows by \(2^2 = 4\) and volume by \(2^3 = 8\).

Answer

Area grows by factor 4; Volume grows by factor 8
Scaling length by \(k\) scales area by \(k^2\) and volume by \(k^3\). Doubling length quadruples area and octuples volume.

About Square vs Cube Intuition

Understanding x2x^2 as the area of a square with side xx (2D), and x3x^3 as the volume of a cube (3D).

Learn more about Square vs Cube Intuition →

More Square vs Cube Intuition Examples

Example 1 easy

A square tile has side length 4 cm. What is its area? A cube has side length 4 cm. What is its volum

Example 3 easy

Find (5^2) and (5^3). What do they represent geometrically?

Example 4 medium

A square garden has area 49 m². A cubic storage box has volume 125 cm³. Find the side length of each

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