Unit 18: Solid Shapes

Volume is area, stacked

A box 4 by 3 by 5 holds 4 × 3 = 12 cubes in its bottom layer, and there are 5 layers. So it holds 60.

That is the whole idea: work out the area of the flat face, then multiply by how tall it is.

It works for any shape that stays the same all the way up — a box, a cylinder, a triangular prism. All of them.

A cylinder is a circle, stacked

A cylinder is exactly the same idea with a round face.

The face is a circle, area pi r². Multiply by the height and you have the volume:

volume = pi r² h

Nothing new was needed. Just a different flat shape at the bottom.

A third for anything with a point

A cone is a cylinder that narrows to a point. A pyramid is a prism that does the same.

Both hold exactly one third of the shape they came from. Fill a cone with water and pour it into a cylinder of the same base and height — it takes three cones to fill it.

So: cone = pi r² h / 3, and pyramid = base area × height / 3.

The sphere

A sphere has no flat face to stack, so it needs its own result:

volume = 4 pi r³ / 3

Archimedes worked this out over two thousand years ago and was so pleased with it that he asked for a sphere inside a cylinder to be carved on his gravestone.

Faces, edges and corners

A solid has faces (the flat sides), edges (where two faces meet) and vertices (the corners).

A cube has 6 faces, 12 edges and 8 vertices.

Counting them carefully is harder than it sounds — the trick is to be systematic: count the top, the bottom, then the sides, rather than jabbing at the picture.

Euler’s astonishing formula

Count the faces, add the vertices, take away the edges. For a cube: 6 + 8 - 12 = 2.

Try a pyramid: 5 + 5 - 8 = 2. A prism: 5 + 6 - 9 = 2.

Always 2. For every solid without a hole in it, however strange. Euler found that in 1750, and it is one of the most surprising facts in all of geometry.

Volume converts by the cube

Length converts by the factor, area by the factor squared, and volume by the factor cubed.

So one cubic metre is 100 × 100 × 100 = 1000000 cubic centimetres. A million.

The pattern is the same each time — the exponent just matches the number of directions.

Litres and cubic centimetres

A litre is defined as 1000 cubic centimetres — a cube 10 cm on each side.

So a 2-litre bottle holds 2000 cm³, and a millilitre is exactly one cubic centimetre.

That is why a medicine spoon marked 5 ml and a cube 5 cm³ hold the same amount. The units were designed to line up.

What shape does a slice make?

Slice a cylinder straight across and the cut face is a circle. Slice it straight down and you get a rectangle.

Slice a cube corner to corner and you can get a triangle — or, cut just right, a perfect hexagon.

A prism is exactly the shape whose cross-section never changes as you slide along it. That is what "prism" means.

A net has to fold up

A net is a solid flattened out. But not every arrangement of squares folds into a cube — some overlap, some leave a gap.

There are exactly eleven different nets for a cube. Not ten, not twelve.

The test is always the same: can you fold it in your head without two faces landing on top of each other?

Filling and emptying

A tank filling at a steady rate is a division problem: volume divided by rate gives the time.

A 60-litre tank filling at 4 litres a minute takes 15 minutes.

The only care needed is that the units match. Litres with litres, minutes with minutes — mixing them is the one thing that goes wrong here.

Density: how heavy for its size

Density is mass divided by volume, and it is why a small lump of lead outweighs a big box of feathers.

Water has a density of 1 gram per cubic centimetre — which is exactly why the gram was defined that way.

So mass equals density times volume, and any two of the three give you the third.

Surface area is a flat problem

Cut a box along its edges and lay it out flat. You get a net — six rectangles.

Surface area is just their total, so it is an ordinary unit 15 problem in disguise.

A box l by w by h has faces in matching pairs: 2(lw + lh + wh).

A cylinder unrolls into a rectangle

Peel the label off a tin and it is a rectangle. Its height is the tin’s height, and its length is the way round the circle, 2 pi r.

Add the two circular ends and you have the lot:

2 pi r h + 2 pi r²

Scaling in three dimensions

Unit 15 said area scales by k^2. Volume scales by k^3, because all three directions grew.

Double a model and it holds 8 times as much, while its surface only grows 4 times. That is why small animals lose heat faster than big ones — and why a giant insect could not breathe.