Volume of a Triangular Prism Formulas - Definition & Examples
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Calculating the Volume of a Triangular Prism A prism is a solid object that has two congruent faces on either end joined by parallelogram faces laterally. A right prism has rectangular faces instead of parallelogram ones. Both these types of prisms have the same formula for volume. The following diagram shows a right triangular prism and its dimensions.
The volume of any prism is equal to the product of its cross section (base) area and its height (length). $V = BA \times l$ In the case of a triangular prism, the base area is the area of the triangular base, which can be calculated using Heronâ€™s formula (if the lengths of the sides of the triangle are known) or by using the standard area of a triangle formula (if the lengths of a side of the triangle and its corresponding altitude are known). $\begin{gathered} Â Â BA = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \hfill \\ Â Â s = \frac{{\left( {a + b + c} \right)}}{2} \hfill \\ \end{gathered} $ We must always take care of the units of measurement in mathematics. For example, if we want the volume in m^{3}, then we need to calculate the base area in ${m^2}$ and the length in $m$. Letâ€™s look at an example to see how to use the formula!
Question: A prism has triangular ends whose sides are 3cm , 4cm and 5cm .If its volume is 84 cm^{3} then find its length. Solution: $\begin{gathered} Â Â a = 3\,cm,\,\,b = 4\,cm,\,\,c = 5\,cm \hfill \\ Â Â s = \frac{{\left( {a + b + c} \right)}}{2} = \frac{{\left( {3 + 4 + 5} \right)}}{2} = \frac{{12}}{2} = 6\,cm \hfill \\ \end{gathered} $ $\begin{gathered} Â Â BA = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \hfill \\ Â Â \quad \,\, = \sqrt {6\left( {6 - 3} \right)\left( {6 - 4} \right)\left( {6 - 5} \right)} \hfill \\ Â Â \quad \,\, = \sqrt {6 \times 3 \times 2 \times 1} = \sqrt {36} = 6\,c{m^2} \hfill \\ \end{gathered} $ $\begin{gathered} Â Â V = BA \times l \hfill \\ Â Â l = \frac{V}{{BA}} = \frac{{84\,c{m^3}}}{{6\,c{m^2}}} = 14\,cm \hfill \\ \end{gathered} $ Why donâ€™t you try to solving a problem yourself to see if you have mastered the formula!
Question: A triangular prism is such that one of the sides of its triangular faces and its corresponding height are equal in length (say x cm). The length and the volume of the prism are respectively equal to 10cm and 80cm^{3}. Find the value of x. Options: (a) 6 cm (b) 4 cm (c) 8 cm (d) none of these Answer: (b) Solution: $\begin{gathered} Â Â BA = \left( {\frac{1}{2}} \right) \times x \times x = \frac{{{x^2}}}{2} \hfill \\ Â Â V = BA \times l \hfill \\ Â Â BA = \frac{V}{l} \hfill \\ Â Â \frac{{{x^2}}}{2} = \frac{{80}}{{10}} = 8 \hfill \\ Â Â {x^2} = 16 \hfill \\ Â Â x = 4\,cm \hfill \\ \end{gathered} $