Here we will learn about the volume of a prism, including how to calculate the volume of a variety of prisms and how to find a missing length given the volume of a prism. Show
There are also volume and surface area of a prism worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. What is the volume of a prism?The volume of a prism is how much space there is inside a prism. Imagine filling this L-shaped prism fully with water. The total amount of water inside the prism would represent the volume of the prism in cubic units.  To calculate the volume of a prism, we find the area of the cross section and multiply it by the depth.
Volume of prism = Area of cross section x depth What is the volume of a prism?How to calculate the volume of a prismIn order to calculate the volume of a prism:
How to calculate the volume of a prismVolume of a prism worksheetGet your free volume of a prism worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Volume of a prism worksheetGet your free volume of a prism worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Volume of a prism examplesExample 1: volume of a triangular prismWork out the volume of the triangular prism:
Volume of prism = Area of cross section × depth 2Calculate the area of the cross section. \[\text{Area of triangle }=\frac{1}{2}\times{b}\times{h}\\ =\frac{1}{2}\times{8}\times{3}\\ =12\] The area of the triangle is 12cm^2 . 3Calculate the volume of the prism. The depth of the prism is 10cm . \[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =12\times{10}\\ =120\] 4Write the answer, including the units. The measurements on this triangular prism are in centimetres so the volume will be measured in cubic centimetres. Volume = 120cm^3 Example 2: volume of a rectangular prism (cuboid)A swimming pool is being built in the shape of a cuboid. Calculate the volume of water in the pool when it is completely filled, in litres. Volume of prism = Area of cross section × depth Calculate the area of the cross section. \[\text{Area of rectangle }=b\times{h}\\ =12\times{3}\\ =36\] The area of the rectangle is 36m^2 . Calculate the volume of the prism. The depth of the prism is 5m . \[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =36\times{5}\\ =180\] Write the answer, including the units. The measurements on this prism are in metres so the volume will be measured in cubic metres, however we then need to convert this to litres, as it is stated in the question. Volume = 180m^3 . 1m^3 = 1000L and so 180m^3 180,000L. The volume of water in the swimming pool in litres is 180,000L . Note: You may also use the formula: Volume of cuboid = height × width × depth since the area of a rectangle is equal to height × width. Example 3: volume of a hexagonal prismWork out the volume of the prism: Volume of prism = Area of cross section × depth Calculate the area of the cross section. In this example, we are told that the area of the hexagon is 50mm^2 so we can move on to the next step. Calculate the volume of the prism. \[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =50\times{15}\\ =750\] Write the answer, including the units. The measurements on this prism are in millimetres so the volume will be measured in cubic millimetres. Volume = 750mm^3 Example 4: volume of a compound prismWork out the volume of the L-shaped prism:
Volume of prism = Area of cross section × depth Calculate the area of the cross section. To calculate the area of the cross section we need to split it into two rectangles and work out the missing side lengths. We can then work out the area of each rectangle: Rectangle A: \[\text{Area }=7\times{4}\\ =28\] Rectangle B: \[\text{Area }=6\times{5}\\ =30\] Total area: 28+30=58cm^2 Calculate the volume of the prism. The depth of the prism is 12cm . \[\text{Volume of prism }=\text{Area of cross section }\times\text{depth}\\ =58\times{12}\\ =696\] Write the answer, including the units. The measurements on this prism are in cm so the volume will be measured in cm^3 . Volume = 696cm^3 Example 5: volume of a trapezoidal prismWork out the volume of the prism:
Volume of prism = Area of cross section × depth Calculate the area of the cross section. \[\text{Area of cross section }=\frac{1}{2}(a+b)h\\ =\frac{1}{2}(2+4) \times 3\\ =9\mathrm{cm}^{2}\] Calculate the volume of the prism. \[\text{Volume of prism }= \text{Area of cross section } \times \text{ depth}\\ =9 \times 5\\ =45\] Write the answer, including the units. Volume = 45cm^3 . How to work out a missing length given the volumeSometimes we might know the volume and some of the measurements of a prism and we might want to work out the other measurements. We can do this by substituting the values that we know into the formula for the volume of a prism and solving the equation that is formed.
Missing length examplesExample 6: missing length in a pentagonal prismThe volume of this prism is 225cm^2 . Work out the depth, L, of the prism: Volume of prism = Area of cross section × depth Calculate the area of the cross section. In this example, we are told the area of the cross section is 25cm^2 Substitute known values into the formula, and solve the equation. \begin{aligned} \text{Volume of prism }&=\text{Area of cross section }\times \text{depth}\\ 225&=25 \times D\\ 25D&=225\\ D&=9 \end{aligned} Write the answer, including the units. Since the units in this question are in cm and cm^3 , the depth of the prism is 9cm . Example 7: missing height in a trapezoidal prismThe volume of this prism is 336cm^3 . Work out the height of the prism. Volume of prism = Area of cross section × depth Calculate the area of the cross section. \[\text{Area of trapezium }=\frac{1}{2}(a+b)h\\ =\frac{1}{2}(5+9) \times h\\ =7h\] Substitute known values into the formula, and solve the equation. \[\text{Volume of prism }=\text{Area of cross section } \times \text{depth}\\ 336=7h \times 8\\ 336=56h\\ 56h=336\\ h=6\] Write the answer, including the units. Since the units in this question are in cm and cm^3 , the height of the prism is 6cm . Common misconceptions
You should always include units in your answer. Remember, volume is measured in units cubed (e.g. mm^3, cm^3, m^3 etc)
You need to make sure all measurements are in the same units before calculating volume. (E.g. you can’t have some in cm and some in m )
Be careful to apply the correct prism related formula to the correct question type. Practice volume of a prism questions\text{Area of triangle }=\frac{1}{2} \times 4 \times 6\\ =12\mathrm{cm}^{2} \text{Volume of triangular prism }=12 \times 8\\ =96\mathrm{cm}^{3} \text{Area of trapezium }=\frac{1}{2}(5+7) \times 4\\ =24\mathrm{cm}^{2} \text{Volume of prism }=24 \times 11\\ =264\mathrm{cm}^{3} Area of cross section = 24cm^2 \text{Volume of prism }=24 \times 10\\ =240 \mathrm{cm}^{3} Area of triangle A : \text{Area }=\frac{1}{2} \times 7 \times 6\\ =21\mathrm{cm}^{2} Area of rectangle B : \text{Area }=7 \times 9\\ =63 \mathrm{cm}^{2} \text{Total area: } 21+63=84\mathrm{cm}^{2} \text{Volume of prism }=84 \times 16\\ =1344\mathrm{cm}^{3} \text{Volume of prism} = \text{Area of cross section} \times \text{depth}\\ 156=12x\\ 12x=156\\ x=13 \text{Area of parallelogram }=5 \times h \begin{aligned} \text{Volume of prism }&=\text{Area of cross section }\times \text{depth}\\ 270&=5h \times 9\\ 270&=45h\\ 45h&=270\\ h&=6 \end{aligned} Volume of a prism GCSE questions1. Work out the volume of the prism. State the units in your solution. (3 marks) Show answer \text{Area of cross section }=2 \times 4 + 4 \times 1=12\text{cm}^2 (1) \text{Volume of prism: }12 \times 5=60 (1) 60cm^3 (1) 2. The volume of the cuboid is twice the volume of the triangular prism. Work out the height, y , of the cuboid.
(5 marks) Show answer \frac{1}{2} \times 9 \times 4=18\mathrm{cm}^{2} (1) 18 \times 5=90 \mathrm{cm}^{3}
(1) 3 \times y \times 6=18y (1) 90 \times 2=180\text{ and } 18y=180 (1) y=10cm (1) 3. (a) Calculate the volume of the trapezoidal prism. (b) The prism is made from aluminum, which has a density of 2.7g/cm^3 . Work out the mass of the prism. State the units in your answer. (4 marks) Show answer (a) (1) 16 \times 8=128 \mathrm{cm}^{3} (1) (b) (1) =345.6g (1) Learning checklistYou have now learned how to:
Still stuck?Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. Find out more about our GCSE maths revision programme. What is the formula for volume prisms?The formula for the volume of a prism is V=Bh , where B is the base area and h is the height. The base of the prism is a rectangle. The length of the rectangle is 9 cm and the width is 7 cm. The area A of a rectangle with length l and width w is A=lw .
What are the two formulas for volume of a prism?Since the cross-section of the triangular prism is a triangle, the formula for the volume of a triangular prism is given as:. The volume of a Triangular Prism = (½) abh cubic units.. The volume of a Rectangular Prism = l.b.h cubic units.. The Volume of a Pentagonal Prism = (5/2) a.b.h cubic units.. |
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