How to Find the Surface Area of a Triangular Prism

A triangular prism has five faces, not six, and only two of them match. That asymmetry is what makes it feel harder than a box, but the method is actually simpler once you stop thinking about faces one at a time. Two ends and one wrapper, and the wrapper is a single rectangle that has been folded twice.

Five faces, and only two of them match

The two triangular ends are identical, so one calculation covers both. The other three faces are rectangles, all of them the same length as the prism, but each as wide as a different side of the triangle. For a 3 by 4 by 5 triangle on a prism 10 long they are 30, 40 and 50.

Add those three rectangles and you get 120, which is also 12 times 10, the perimeter of the triangle times the length. That is not a coincidence, it is the whole point: the three rectangles are one strip wrapped around the triangle, and the strip is as long as the way round. So the calculation collapses to two triangles plus one wrap.

The net makes it obvious

Unfold the prism and the three rectangles land side by side in a strip, with a triangle stuck on each end. Nothing about that picture needs a formula. Every rectangle is length times width, every triangle is half base times height, and the total is the surface area.

A worked check you can repeat on paper

That last row is there as a warning. A triangle has three different heights, and 2.4 is the one measured to the hypotenuse. None of them belongs in the rectangles. The rectangles are as wide as the triangle’s sides, never as wide as its heights, and mixing the two is the fastest way to a wrong answer that still looks plausible.

When the ends are given as three sides

Textbook prisms have right angled ends, so the area is half of two legs multiplied. Real ones often do not, and you get three side lengths and no height at all. Heron handles that without complaint: half the perimeter, then multiply s by s minus each side, then take the square root.

Take a prism 10 long whose ends have sides of 13, 14 and 15. The semi perimeter is 21, so the area is √(21 × 8 × 7 × 6), which is √7056, which is 84. Two ends make 168. The perimeter is 42, so the wrap is 420. The surface area is 588. Notice that the wrap is far bigger than the ends here, which is normal for anything long and thin, and it is why the perimeter matters more than the end area on a long prism.

Mistakes that catch people out

• Using a height where a side belongs. The rectangles are as wide as the triangle’s sides. In the worked example the sides are 3, 4 and 5, and the heights are 4, 3 and 2.4. Feeding 2.4 into a rectangle instead of 5 costs 26 square units.
• Forgetting the third rectangle. Two of the three are usually visible in a drawing and the third is round the back. Every side of the triangle gets a rectangle, always three of them.
• Assuming the rectangles are equal. They only match if the triangle is equilateral. For a 3 by 4 by 5 end they are 30, 40 and 50, which are not close.
• Doubling the wrap as well as the ends. The 2 applies only to the triangular ends. The wrap already goes the whole way round, so doubling it counts every side twice.
• Using a slant length instead of the prism length. If the prism is drawn in perspective, the length shown on the page is not the length to use. Take it from the dimension line, not the drawing.

Try these

Common questions

What is the formula for the surface area of a triangular prism?

Twice the area of the triangular end, plus the perimeter of that triangle multiplied by the length of the prism. For 3 by 4 by 5 ends on a prism 10 long that is 2 times 6 plus 12 times 10, which is 132 square units.

How many faces does a triangular prism have?

Five. Two triangular ends that are identical to each other, and three rectangles, one for each side of the triangle. Only the two ends match, so the three rectangles have to be worked out separately unless the triangle is equilateral.

How do you find the area of the triangular end?

Half the base times the perpendicular height if you have a height. If you only have the three side lengths, use Heron: halve the perimeter to get s, then take the square root of s times s minus each side in turn.

Why is the wrap the perimeter times the length?

Because the three rectangles are all the same length as the prism, and their widths are the three sides of the triangle. Adding those widths gives the perimeter, so the three rectangles together are perimeter times length.

Does this formula work for other prisms?

Yes, for any prism at all. Twice the area of the end plus the perimeter of the end times the length works for hexagonal prisms, L shaped extrusions and boxes alike. It is the general rule, and the box formula is just this one with a rectangular end.

What units does surface area use?

Square units, because every face is one length multiplied by another. If your answer came out in cubic units you have calculated the volume instead, which for this prism would be 6 times 10, or 60.

Related lessons and calculators

• Surface area of a rectangular prism, the same two ends and a wrap idea with a rectangle at the end instead of a triangle.
• Area of a triangle from three sides, the Heron step above, worked through properly.
• Area of a triangle, for choosing between the four ways of measuring the end.
• Area formulas for every common shape, since a prism is only ever as hard as the shape on its end.
• Volume of 3D shapes calculator, for the other question about the same solid.
• the National Council of Teachers of Mathematics, for how nets and solids are sequenced in the curriculum.
• Surface area of a cone, the same unfolding idea when the sides curve instead of staying flat.
• Volume of a triangular prism, the same solid measured for what it holds rather than for what it takes to cover it.