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This page sets out the method for volume of a prism in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Volume of a prism topic page.
When the solid has the same shape all the way through, find the end face area first.
When the volume is given and a length is missing.
A prism has the same cross-section all the way along.
Find the area of that shape first.
Volume = area of cross-section × length.
The length is perpendicular to the cross-section.
The cross-section is a rectangle, so area = length × width, and volume = that × height.
The three-length product is the special case.
The cross-section is a triangle, so use half of base times height.
Then multiply by the length of the prism.
1 litre = 1000 cm³.
Convert if the question asks for litres.
If the volume and the cross-section area are known, divide to find the length.
Length = volume ÷ area.
Do the area of the cross-section as its own sum. Write the area, with cm^2, before you multiply by the length. For a rectangle the area is length times width. For a triangle it is 1/2 × base × height. For a trapezium it is the mean of the parallel sides times the distance between them. Only after that number is on the page do you multiply by how far the prism stretches.
The length has to be perpendicular to the cross-section. On a triangular prism the “length” is often the long edge that is not a side of the triangle. Read the diagram twice: which lengths belong to the end, and which length runs along the solid. Using a side of the triangle as the length, and forgetting to find the area, produces a number that is not the volume.
Method: 1. Find the cross-section area first and write it in cm^2. 2. Then multiply that area by the length perpendicular to the cross-section. 3. Use 1/2×base×height for a triangular end.
A trapezium has one pair of parallel sides. Its area is (a + b)/2 × h, where a and b are the parallel sides and h is the perpendicular distance between them. For parallel sides 8 cm and 4 cm and height 5 cm, the mean of the parallel sides is (8 + 4)/2 = 6 cm. Times the height, the area is 6 × 5 = 30 cm^2.
The prism length is a separate measurement. If that prism is 15 cm long, the volume is 30 × 15 = 450 cm^3. Do not add 8 + 4 + 5 + 15 and call it a volume. Do not multiply 8 × 4 × 5 × 15. The 8 and the 4 are combined by adding and then halving, because they are the two parallel sides of one face. The 5 is the height of that face. The 15 is the only length that runs along the prism.
Method: 1. Average the parallel sides, then multiply by the perpendicular height. 2. Sides 8cm and 4cm with height 5cm give area (8+4)/2× 5=30cm^2. 3. A prism length of 15cm gives volume 30× 15=450cm^3.
Real exam questions often give you the volume and ask for a missing length. The formula V = A × l still works, but you rearrange it. Divide the volume by the length of the prism to get the area of the cross-section. Then use the area formula for the shape to find the missing side.
A typical set-up is a triangular prism with the base, the length and the volume given. Divide the volume by the length to get the area, for example 480 ÷ 20 = 24 cm^2. Then solve 1/2 × 6 × h = 24. Do the two stages on separate lines so each method mark is easy to see.
Method: 1. Write V = A × l and put in the volume and the length. 2. Divide the volume by the length to get the cross-section area. 3. Put that area into the area formula for the cross-section and solve for the missing side. 4. Check by working forwards: area times length must give the volume you were given.
Tip: The first division gives an area, never a length. Write cm^2 next to it so you do not mix them up.
Not every cross-section is a single triangle or rectangle. Exam questions use L-shapes, rectangles with a corner removed, and a trapezium that is easier to split. Find the area of the whole cross-section first. You can split it into rectangles and add, or take a small rectangle away from a large one.
Once you have the single cross-section area, the rest is the same: multiply by the length. Mark the lengths you have used on the sketch. A very common slip is to forget that some sides of an L-shape are not given directly and must be found by adding or subtracting the sides you know.
Method: 1. Sketch the cross-section and decide whether to add pieces or subtract a piece. 2. Work out each piece area and write the units as cm^2. 3. Add or subtract to get the total cross-section area. 4. Multiply by the length of the prism and write the answer in cm^3.
Volume questions often finish with a unit change or a second step. One litre is 1000 cm^3, so a volume in cm^3 becomes litres when you divide by 1000. One cubic metre is 1 000 000 cm^3, so convert lengths to the same unit before you multiply, not afterwards.
For a solid made of a material, the mass is the density times the volume. With density in g/cm^3 and volume in cm^3 the mass comes out in grams. Divide by 1000 for kilograms. Cost questions work the same way: volume times the price per unit of volume.
Method: 1. Check every length is in the same unit before multiplying. 2. Find the volume in cm^3 (or m^3 if every length is in metres). 3. Convert: divide by 1000 for litres, or use mass = density × volume. 4. Write the final unit clearly: litres, grams, kilograms or pounds.
Tip: Density in g/cm^3 needs a volume in cm^3. If the lengths are in metres, change them first.
Some prism questions are not about volume at all. A prism with an n-sided cross-section has n + 2 faces, 3n edges and 2n vertices. A triangular prism has 5 faces and 9 edges. A hexagonal prism has 8 faces and 18 edges. Learn the pattern rather than counting from a picture.
For surface area, add up every face. The two ends are the cross-section, so double its area. The rectangles form one long strip whose width is the perimeter of the cross-section and whose length is the length of the prism. So the total is 2A + P × l, which saves finding each rectangle separately.
Method: 1. Count the sides of the cross-section, n, then write faces = n + 2. 2. Find the area of one end and double it. 3. Find the perimeter of the cross-section and multiply by the length. 4. Add the two parts and write the answer in cm^2.
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