Cube

Properties

• 6 faces, 12 edges and 8 corners.
• Every face is a square and every angle is a right angle.
• The lateral surface is the four side faces, leaving out the top and the bottom.
• The space diagonal runs from one corner through the centre to the opposite corner.

Why the formulas work

A cube of edge 4 cm holds 4 layers, each of 4 rows of 4 unit cubes: 4 × 4 × 4 = 64. So the volume is the edge cubed. Each of the six faces is a square of area a², giving 6a² for the whole surface. The space diagonal uses Pythagoras twice: first across a face (a√2), then up through the cube, giving a√3.

Worked example

Take a cube with edge 5 cm.

1. Volume. Volume = a³. Here: 5 × 5 × 5 = 125 cm³.
2. Total surface area. Total surface area = 6a². Here: 6 × 5 × 5 = 150 cm².
3. Lateral surface area. Lateral surface area = 4a². Here: 4 × 5 × 5 = 100 cm².
4. Space diagonal. Diagonal = a√3. Here: 5 × √3 = 8.66 cm.

Finding a missing measurement

From the volume, the edge is the cube root: 343 cm³ gives an edge of 7 cm. From the total surface area, a = √(T ÷ 6): 150 cm² gives a² = 25, so a = 5 cm and the volume is 125 cm³.

Mistakes to avoid

• Doubling the edge makes the volume eight times as large, not double.
• Giving volume in square units. Volume is in cubic units (cm³).
• Using 4a² when the question asks for the total surface.

Where you see it

Dice, sugar cubes, ice cubes, a puzzle cube and some packing boxes are cubes.

Try these

Work each one out, then tap to check.

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