Area and volume formulae

You do not need a brand new formula for every solid. Lots of volume formulas share one simple idea.

For any prism, the volume is the base area times the height:

Volume of a prism = base area x height

The height is always measured straight up, at a right angle to the base. That stays true even if the prism is lying on its side or leaning over.

Now watch the same idea appear again and again. A rectangular prism has a rectangle for a base, so its volume is length x width x height. A triangular prism has a triangle for a base, so its volume is half x base x height of the triangle, times the length of the prism. A cylinder works like a prism too, but its base is a circle with area pi x r squared, so its volume is pi x r squared x h.

Once you see that every one of these is just base area times height, the whole family of formulas is much easier to remember.

Cavalieri's principle says: if two solids have the same height, and every slice across them at the same level has the same area, then the two solids have the same volume.

Think of a neat stack of coins and a slanted stack of the same coins. The slanted stack looks messy, but each coin is still there, so the stacks take up the same amount of space.

This is why the prism rule works even for tilted solids. A leaning cylinder and a straight cylinder with the same base and height hold the same volume, because every matching slice has the same area. It also explains why we can trust the formula volume = cross-sectional area x height: only the slice areas matter, not how straight or slanted the solid looks.

A prism is a solid with two identical ends, called bases, joined by flat sides. Think of a Toblerone bar or a tent: the same shape runs all the way through.

To find the volume of any prism, use two steps. First, work out the area of one base. Second, multiply that area by the length between the two bases.

For a triangular prism, the base is a triangle, so its area is half x base x height of the triangle. Multiply that by the length of the prism and you are done. For example, if the triangle has base 6 cm and height 4 cm, its area is 12 square centimetres. If the prism is 10 cm long, the volume is 12 x 10 = 120 cubic centimetres.

The same two steps work for every prism. Find the base area, then multiply by the length.

Some solids are mash-ups of shapes you already know. To find their volume, split them into parts you recognise, like prisms and cylinders, and use the formulas for those parts.

If the solid is built by joining shapes together, add the volumes. A model house made of a box with a triangular prism roof has a volume equal to the box volume plus the roof volume.

If the solid has a chunk missing, subtract the volumes. A block with a cylindrical hole drilled through it has a volume equal to the block volume minus the cylinder volume.

The trick is to spot the familiar shapes hiding inside the big one. Once you have split it up, each part is just base area times height, and you already know how to do that.