Liri had found a small cuboid-shaped box full of stardust. “How much stardust actually fits in there?”
Erklärus: “That is the volume. For a cube and a cuboid the formula is very simple – length times width times height.”
How much fits inside?
🔷 Properties – what makes a cube and a cuboid
Both solids belong to the same family: the base and top face are parallel and identical, and both have exactly 8 vertices, 12 edges and 6 faces.
Key point: “A cube is a special case of the cuboid – just as the square was a special case of the rectangle. Whenever a = b = c, the cuboid becomes a cube.”
📐 The surface area
The surface area is the sum of all the faces on the outside.
Liri: “If I wanted to paint a box like that, how much paint would I need?”
Erklärus: “Exactly the right question – that is the surface area. Imagine you cut the box open along its edges and fold it flat. Then you see its net.”
Six squares, unfolded – each of them a² in size.
Erklärus: “With a cube all six faces are the same size – each a square with area a². With a cuboid, two opposite faces are always the same size, but the three pairs of faces differ from one another.”
Formula – surface area
SA(cube) = 6 × a²
SA(cuboid) = 2 × (a×b + a×c + b×c)
Cube with edge 3 cm – surface area?
📦 Working out the volume
In a cube all the edges are the same length. A cuboid has a length, a width and a height.
Liri: “And how much space is there in a box like that? Like the area of the tile, only now in three directions?”
Erklärus: “Exactly the same idea! Imagine we build a cuboid out of lots of little unit cubes – each exactly 1 cm on every side. How many fit in?”
3 unit cubes long · 2 deep · 2 high = 3 × 2 × 2 = 12 unit cubes
Liri: “Three side by side, times one row deep, times two layers on top of each other… that is length times width times height again!”
Erklärus: “Exactly the same principle as with area – only now with a third direction. The volume counts how many unit cubes fit in without any gaps.”
Formula – volume
V(cube) = a × a × a = a³
V(cuboid) = a × b × c
Volume is measured in cubic centimetres (cm³) or cubic metres (m³).
Cube with edge 4 cm – volume?
🌌 Spaces in the universe
Many satellites are built in the shape of a cuboid – that can be stowed away most efficiently in a rocket.
A small research satellite measuring 1 m × 1 m × 1 m has a volume of exactly 1 m³.
Without calculating volume, engineers couldn’t know how much equipment fits into a satellite.
❌ Common mistake
Volume (cm³) and surface area (cm²) are often mixed up – and so are the formulas!
Tip: volume = how much fits inside (3 measurements multiplied). Surface area = how much is covered on the outside (faces added).
📦 Exercise 1 – Volume of a cuboid
Cuboid: 5 cm × 2 cm × 3 cm. What is the volume?
🎲 Exercise 2 – Volume of a cube
Cube with edge 5 cm. What is the volume?
📐 Exercise 3 – Surface area
Cube with edge 2 cm. What is the surface area?
The box is measured
Liri now knew exactly how much stardust fitted into her box – and how much material would be needed to build a new one.
Liri: “Now I can even work out how big my home would have to be to pack in everything I collected on the way.”
“Volume: V(cube)=a×a×a, V(cuboid)=a×b×c, unit cm³. Surface area: SA(cube)=6×a×a, unit cm².”
What you learned today
V = a×b×c · SA(cube) = 6×a²
Volume in cm³/m³ · Surface area in cm²/m²
🌟
Chapter 29 mastered!
Liri can now work out the volume and surface area of cubes and cuboids.
Erklärus’s notebook – page 29:
“Cube: V=a×a×a, SA=6×a×a. Cuboid: V=a×b×c. Volume measures how much fits inside (cm³). Surface area measures how much is covered on the outside (cm²).”
“Cube: V=a×a×a, SA=6×a×a. Cuboid: V=a×b×c. Volume measures how much fits inside (cm³). Surface area measures how much is covered on the outside (cm²).”