Cuboid

Properties

• 6 faces, 12 edges and 8 corners.
• Opposite faces are equal rectangles.
• The lateral surface is the four walls, without the floor and the ceiling.
• The longest rod that fits inside is the space diagonal.

Why the formulas work

The floor of the box holds l × b unit cubes in one layer, and there are h layers, so the volume is l × b × h. The surface is three pairs of equal rectangles: top and bottom (lb each), front and back (lh each), and the two sides (bh each). The four walls alone have area equal to the perimeter of the floor times the height.

Worked example

Take a cuboid with length 12 cm, breadth 8 cm and height 5 cm.

1. Volume. Volume = l × b × h. Here: 12 × 8 × 5 = 480 cm³.
2. Total surface area. Total surface area = 2(lb + bh + hl). Here: 2 × (12 × 8 + 8 × 5 + 5 × 12) = 392 cm².
3. Lateral surface area. Lateral surface area = 2h(l + b). Here: 2 × 5 × (12 + 8) = 200 cm².
4. Space diagonal. Diagonal = √(l² + b² + h²). Here: √(12² + 8² + 5²) = 15.26 cm.

Finding a missing measurement

Divide the volume by the two measures you know: a volume of 480 cm³ with length 12 cm and breadth 8 cm gives a height of 480 ÷ 96 = 5 cm. For a tank, the depth of water is the volume poured in ÷ the area of the base: 240 litres, which is 2,40,000 cm³, in a tank with a base of 80 cm by 60 cm stand 2,40,000 ÷ 4800 = 50 cm deep.

Mistakes to avoid

• Using different units for the three measures. Change everything to the same unit first.
• For painting the walls of a room, use the lateral surface area and then take away doors and windows.
• A litre is 1000 cm³, so a tank 50 cm × 40 cm × 30 cm holds 60 litres.

Where you see it

Bricks, books, rooms, water tanks, shipping containers, match boxes and almirahs are cuboids.

Try these

Work each one out, then tap to check.

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