Volume of Prisms
What a right prism is
A right prism is a solid with a uniform cross-section: the same flat shape runs all the way along its length without changing. A box, a triangular wedge, and an L-shaped bar of metal are all prisms, because a single slice describes the whole solid. The cross-section is the shape you see when you look at the prism end-on, straight at one end. The word right means the sides stand square to the two ends, so the length is measured straight across rather than on a slant. Learning to spot the cross-section first is the habit that makes every volume question below straightforward.
Volume is cross-section times length
The central idea of this unit is one rule: the volume of a right prism is the area of its cross-section times its length, written V = A × L. Picture the cross-section copied again and again, stacked all along the length like a deck of identical cards; the area of one card times how far the stack runs gives the space inside. This single rule covers every right prism, whatever shape the cross-section happens to be. For example, if the cross-section has area A = 12 cm² and the length is L = 5, the volume is 12 × 5 = 60 cm³.
The volume of a rectangular prism
A rectangular prism, also called a cuboid, has a rectangle as its cross-section. Because the area of a rectangle is length × width, the rule V = A × L turns into the familiar V = length × width × height. Take a box measuring 5 by 4 by 3 cm. Its base is a 5 × 4 rectangle, so the base area is 5 × 4 = 20 cm², and that base is the cross-section. Multiplying by the height of 3 gives V = 20 × 3 = 60 cm³, which is exactly the same as computing 5 × 4 × 3 in a single line.
The volume of a triangular prism
A triangular prism has a triangle as its cross-section. The area of a triangle is (1/2) × base × height, so the reliable plan is to find that area first and then multiply by the length of the prism. Suppose the triangle has base 6 and height 4; its area is (1/2) × 6 × 4 = 12 cm². If the prism is 10 long, the volume is 12 × 10 = 120 cm³. The order of steps matters for clear thinking: work out the cross-section area first, then times the length. The same plan handles any prism once you can measure its end face.
From volume to capacity
Capacity is the amount a container can hold, and it is tied directly to volume. Two unit links are worth memorising: 1 cm³ = 1 mL, and 1000 cm³ = 1 L, which also means 1000 mL = 1 L. So once you know a volume in cubic centimetres, the capacity follows at once. A container with a volume of 2000 cm³ holds 2000 mL, which is the same as 2 L. Choosing sensible units matters as well: use cm³ or m³ for the volume of a solid, and mL or L for the capacity it can hold.
Why this matters
Volume and capacity show up everywhere: packing boxes, filling tanks and pools, pouring drinks, and building. The single idea of cross-section area times length handles every right prism, so a wedge, a length of guttering, and a plain box all give way to the same method. The links 1 cm³ = 1 mL and 1000 cm³ = 1 L connect the size of a solid to what it can hold, which is why a fish tank measured in centimetres can be quoted in litres. The usual pitfall is units: volume is counted in cubic units, while capacity is counted in mL or L. This unit stays with right prisms; circles and cylinders, which need pi, come in a later unit.