Triangular prism – surface area and volume
Type the legs of the base and the height of the prism – the surface area and volume are calculated automatically. Below are three methods, including “half a box”.
Three methods, one result
Method 1 · Face by face
First the third side of the base, from Pythagoras’ theorem: c² = 3² + 4² = 25, c = 5.
Two bases (triangles): 2 · 3 · 4 : 2 = 12
Side faces: 3 · 5 = 15, 4 · 5 = 20, 5 · 5 = 25
Pc = 12 + 15 + 20 + 25 = 72
Method 2 · Wrap-around strip
The side faces unrolled into one rectangle: perimeter of the base · height.
Perimeter of the base: 3 + 4 + 5 = 12; strip: 12 · 5 = 60
The two bases together make a rectangle 3 × 4 = 12; Pc = 60 + 12 = 72
Method 3 · Half a box
A cuboid 3 × 4 × 5 cut vertically along the diagonal of its base gives two such prisms.
Surface area of the box: 2 · (12 + 15 + 20) = 94; each prism gets half: one triangle at the bottom and one at the top (together 12), one face 3 × 5 and one 4 × 5, that is 47.
Plus the new face made by the cut: 5 · 5 = 25. Pc = 47 + 25 = 72
Volume: area of the base · height = 3 · 4 : 2 · 5 = 6 · 5 = 30.
Check with half a box: 3 · 4 · 5 = 60, half of it is 30.
Watch out: the area of a triangular base is a · b : 2. Without dividing by 2 you get the area of a rectangle, which is twice the correct value.
How do you work out the surface area of a triangular prism?
A right triangular prism (graniastosłup trójkątny, more precisely graniastosłup prosty) has two bases (podstawa) and side faces (ściany boczne). Here the base is a right-angled triangle (trójkąt prostokątny) with legs (przyprostokątne) a and b, so the area of the base (pole podstawy) is a · b : 2, and the two bases together make a rectangle a × b. The side faces are three rectangles: a × H, b × H and c × H, where c is the third side of the base, the hypotenuse (przeciwprostokątna). Together they make the lateral surface area (pole powierzchni bocznej).
You get the third side from Pythagoras’ theorem (twierdzenie Pitagorasa): c² = a² + b². For a = 3 and b = 4: c² = 9 + 16 = 25, c = 5. The total surface area (pole powierzchni całkowitej, Pc) with H = 5: Pc = 12 + 15 + 20 + 25 = 72.
In Polish textbooks and in the exam the total surface area is usually written Pc, the volume (objętość) V and the height of the prism (wysokość) H. A raised dot · means multiplication, a colon : means division, and decimals are written with a comma: 2,5 rather than 2.5. The sign × is used only between dimensions, as in a 3 × 4 × 5 box.
How do you work out the volume of a prism?
The volume of any right prism is V = area of the base · height. For a base of 3 and 4: area of the base = 3 · 4 : 2 = 6, and with H = 5: V = 6 · 5 = 30. That is half of a 3 × 4 × 5 box, a cuboid (prostopadłościan) with a volume of 60.
When do pupils in Poland learn about prisms?
The volume and surface area of right prisms and Pythagoras’ theorem are in the curriculum for grades 7–8 (klasy VII–VIII) of Polish primary school (8-year szkoła podstawowa; podstawa programowa, Dz.U. 2024 poz. 996). The eighth-grade exam (egzamin ósmoklasisty) in 2025 and 2026 included problems about solids (bryły) (CKE report 2025, CKE report 2026; documents in Polish).
What is the most common mistake?
Working out the area of the triangular base as if it were a rectangle, without dividing by 2. The volume then comes out twice as big as it should be. Another common mistake is leaving out the third side face (the one along the hypotenuse), because the hypotenuse c has to be worked out first.
Mini glossary: English – Polish
- right prism – graniastosłup prosty
- triangular prism – graniastosłup trójkątny
- solid – bryła
- base – podstawa
- area of the base – pole podstawy
- side face – ściana boczna
- lateral surface area – pole powierzchni bocznej
- total surface area – pole powierzchni całkowitej (Pc)
- volume – objętość (V)
- height – wysokość (H)
- right-angled triangle – trójkąt prostokątny
- leg (of a right-angled triangle) – przyprostokątna
- hypotenuse – przeciwprostokątna
- perimeter – obwód
- cuboid – prostopadłościan
- Pythagoras’ theorem – twierdzenie Pitagorasa
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