Practice Square vs Cube Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

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

52=255^2 = 25 is a 5×55 \times 5 square's area. 53=1255^3 = 125 is a 5×5×55 \times 5 \times 5 cube's volume.

Showing a random 20 of 50 problems.

Example 1

easy
A square has side 11. What is its area?

Example 2

easy
Compute 232^3.

Example 3

medium
A square's side triples from 22 to 66. By what factor does its area change?

Example 4

hard
A square has area xx and a cube has volume xx, both with the same number xx. The square's side equals the cube's edge. Find all positive xx.

Example 5

medium
A storage box is a cube with edge 0.50.5 m. How many cubic centimetres of stuff does it hold?

Example 6

easy
Which is larger, 424^2 or 434^3?

Example 7

easy
Compute 626^2 and state what shape it represents.

Example 8

easy
A square has side length 44. What is its area?

Example 9

challenge
Two cubes have edges aa and 2a2a. Show by what factor the larger cube's volume exceeds the smaller, for any aa.

Example 10

easy
A square's area is 3636. What is its side length?

Example 11

easy
A cube has volume 216216 cm³. Find its edge.

Example 12

challenge
A cube and a square share the same numeric measure: a cube of edge xx has volume equal to the area of a square of side xx. Find all positive xx.

Example 13

easy
A cube has edge 44 cm. Find its volume.

Example 14

easy
A square has side 77. Find its area.

Example 15

hard
A cube has total surface area 150150 cm². Find its volume.

Example 16

challenge
A cube of edge aa and a cube of edge bb have a combined volume equal to a single cube of edge cc. If a=3,b=4a=3, b=4, find cc (to 3 decimal places).

Example 17

easy
Compute 333^3 and state what shape it represents.

Example 18

medium
Which is larger: the area of a square of side 1010, or the volume of a cube of edge 44?

Example 19

medium
A square tile has area 4949 and a cube has volume 6464. Which has the larger side/edge length?

Example 20

medium
A square's side length doubles from 33 to 66. By what factor does its area change?
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