ACARA v9 CONTENT DESCRIPTION “solve problems involving the volume and capacity of right prisms using appropriate units”
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.
A right prism has one cross-section
The same end shape runs all along the length.
a right prism has the same cross-section all along its length; the cross-section is the shape you see end-on.
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 x 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 cm^2 and the length is L = 5, the volume is 12 x 5 = 60 cm^3.
Volume = cross-section area x length
Stack the cross-section along the length: V = A x L.
the volume of any right prism is the area of its cross-section times its length: V = A x L.
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 x width, the rule V = A x L turns into the familiar V = length x width x height. Take a box measuring 5 by 4 by 3 cm. Its base is a 5 x 4 rectangle, so the base area is 5 x 4 = 20 cm^2, and that base is the cross-section. Multiplying by the height of 3 gives V = 20 x 3 = 60 cm^3, which is exactly the same as computing 5 x 4 x 3 in a single line.
A rectangular prism
Base area times height: 5 x 4 x 3.
for a rectangular prism, V = length x width x height; here 5 x 4 x 3 = 60 cm^3.
The volume of a triangular prism
A triangular prism has a triangle as its cross-section. The area of a triangle is (1/2) x base x 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) x 6 x 4 = 12 cm^2. If the prism is 10 long, the volume is 12 x 10 = 120 cm^3. 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.
A triangular prism
Triangle area first, then times the length.
for a triangular prism, find the triangle area then times the length: (1/2 x 6 x 4) x 10 = 120 cm^3.
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 cm^3 = 1 mL, and 1000 cm^3 = 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 cm^3 holds 2000 mL, which is the same as 2 L. Choosing sensible units matters as well: use cm^3 or m^3 for the volume of a solid, and mL or L for the capacity it can hold.
From volume to capacity
1 cm^3 = 1 mL and 1000 cm^3 = 1 L.
capacity is what a container holds; 1 cm^3 = 1 mL and 1000 cm^3 = 1 L, so a 2000 cm^3 box holds 2 L.
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 cm^3 = 1 mL and 1000 cm^3 = 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.
Quick self-check
1. What is the volume of a rectangular prism measuring 5 x 4 x 3 cm?
2. The volume of any right prism equals...
3. A triangular prism has a triangular face of area 12 cm^2 and length 10 cm. What is its volume?
4. How many millilitres is 1 cm^3?
5. A container has a volume of 2000 cm^3. What is its capacity in litres?