Surface area of a triangular prism
Open the prism conceptually into a net. Count the two triangular ends and three rectangular side faces, using a matching triangle side as the width of each rectangle
\[\begin{gathered}SA=2\left(\frac{1}{2}ab\right)\\{}+L(a+b+c)\end{gathered}\]
- Count every face A triangular prism has two congruent triangular ends and one rectangular face for each side of the triangle \(\displaystyle 2\text{ triangles}+3\text{ rectangles}\)
- Find the triangular ends Use one-half of base times perpendicular height for each triangular end \(\displaystyle \begin{gathered}A_{\text{ends}}=2\left(\frac12\times6\times8\right)\\{}=48\text{ cm}^2\end{gathered}\)
- Match each side Each rectangular face uses the corresponding triangle side and the prism length \(\displaystyle A_c=cL=10\times15=150\text{ cm}^2\)
- Add square areas Add every face area once and give the result in square units \(\displaystyle \begin{gathered}SA=48+90+120+150\\{}=408\text{ cm}^2\end{gathered}\)
Worked example
A right-triangular prism has perpendicular triangle sides 5 cm and 12 cm, hypotenuse 13 cm, and prism length 14 cm. Find its total surface area
\[\begin{gathered}a=5\text{ cm},\quad b=12\text{ cm}\\c=13\text{ cm},\quad L=14\text{ cm}\end{gathered}\]
- \(\displaystyle \begin{gathered}A_{\text{ends}}=2\left(\frac12\times5\times12\right)\\{}=60\text{ cm}^2\end{gathered}\)The two triangular ends have a combined area of 60 cm²
- \(\displaystyle \begin{gathered}A_a=5\times14=70\text{ cm}^2\\A_b=12\times14=168\text{ cm}^2\\A_c=13\times14=182\text{ cm}^2\end{gathered}\)Multiply the prism length by each corresponding triangle side to find the three rectangular-face areas
- \(\displaystyle \begin{gathered}SA=60+70+168+182\\{}=480\text{ cm}^2\end{gathered}\)Add the two-end area and the three rectangular-face areas to obtain the total surface area
Common mistakes
Using the perpendicular height for a side face
✕\(\displaystyle \begin{gathered}SA=60+70+168+168\\{}=466\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=60+70+168+182\\{}=480\text{ cm}^2\end{gathered}\)
Each rectangular face uses its matching triangle side as its width, so the hypotenuse gives 182 cm² for its faceLeaving out a rectangular face
✕\(\displaystyle \begin{gathered}SA=60+70+168\\{}=298\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=60+70+168+182\\{}=480\text{ cm}^2\end{gathered}\)
The net has one rectangular face for each of the three triangle sides, so all three must be includedConfusing surface area and volume
✕\(\displaystyle \begin{gathered}V=\frac12\times5\times12\times14\\{}=420\text{ cm}^3\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=60+70+168+182\\{}=480\text{ cm}^2\end{gathered}\)
Surface area adds face areas in square units, whereas the product of cross-sectional area and prism length gives volume in cubic unitsDifficulty levels
introAdd the faces of a triangular prism when all three triangle sides and the prism length are given
standardUse Pythagoras to find a missing hypotenuse before finding the face areas
challengeHandle an irrational missing side or a mixed-unit measurement before rounding the surface area
examChoose the surface-area value from options containing face, height and volume errors
Questions about this topic
Why is the hypotenuse used for one rectangular face?+
Every rectangular side face has one side of the triangular end as its width, so the face beside the hypotenuse uses the hypotenuse
How is a missing triangle side found?+
Use Pythagoras' theorem, adding the squared perpendicular sides for a missing hypotenuse or subtracting a squared leg from the squared hypotenuse for a missing leg
Why are surface-area units squared?+
Each face is a two-dimensional region, so its area is measured in square units such as cm²
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)