The volume of a prism is cross-section area × length. The cross-section is the shape you see when you look at one end, and the length is the distance between the two identical ends.
This appears in every volume question, and it builds on the area skills in area, perimeter and surface area. It is the first lesson in volume and capacity.
Why does cross-section times length work?
Picture a stack of identical sheets of card, each shaped like the end face. The stack’s volume is the area of one sheet multiplied by how many sheets fit, which becomes the length.
So the job splits into two smaller ones. First find the area of the end shape. Then multiply by the length that runs the other way.
How to work through a prism question
- Identify the two identical end faces. This is your cross-section.
- Find its area with the right formula. For a trapezium, area = ½ × (a + b) × h, where a and b are the parallel sides.
- Identify the length, the distance between the two ends.
- Multiply area by length.
- State the unit as cubic, such as cm³.
Worked example
A prism has a trapezium as its cross-section.
The parallel sides are 7 cm and 10 cm, the perpendicular distance between them is 4 cm, and the slanting side is 5 cm. The prism is 15 cm long. Find its volume.
Step 1, cross-section: the trapezium.
Step 2, area: ½ × (7 + 10) × 4 = ½ × 17 × 4 = 34 cm².
Step 3, length: 15 cm.
Step 4, volume: 34 × 15 = 510 cm³.
The slanting side of 5 cm was given but not needed. Questions often include a measurement you should leave alone.
The mistake to watch for
A common slip is to use the slanting side as the height of the trapezium.
Mistaken working: ½ × (7 + 10) × 5 = 42.5, then 42.5 × 15 = 637.5 cm³
The 5 cm side is slanted, so it is not the perpendicular height.
The correction is to ask whether the measurement makes a right angle with the parallel sides. Here only the 4 cm does, so the answer is 510 cm³. A small right-angle mark on your sketch is a good habit.
Check yourself
Try these, then open each answer.
1. A triangular prism has a right-angled triangle as its end, with base 8 cm and perpendicular height 6 cm. The prism is 10 cm long. Find the volume.
Show answer
Area = ½ × 8 × 6 = 24 cm². Volume = 24 × 10 = 240 cm³.
2. The end face of a prism is a 5 cm by 4 cm rectangle with a 2 cm by 2 cm square cut from one corner. The prism is 9 cm long. Find the volume.
Show answer
Area = 5 × 4 − 2 × 2 = 20 − 4 = 16 cm². Volume = 16 × 9 = 144 cm³.
3. A concrete step is a prism with cross-section area 2.5 m² and length 6 m. What is its volume?
Show answer
Volume = 2.5 × 6 = 15 m³.
Where this leads next
The next step is working with a cylinder and a cone, where the cross-section becomes a circle. The non-calculator working trainer is handy for practising the arithmetic.
Some students can copy a formula but cannot choose the right cross-section in a new diagram. That is something teachers can spot in a few minutes, and it is part of what online one-to-one Mathematics tuition looks at.