acalculator

How big is my triangular prism?

Type the size of the triangle at the end of the prism and the prism's length. The triangular prism calculator gives the volume, the base area and the surface area.

Your numbers

Triangle known by
Volume
60

A triangular prism 10 long with a base area of 6 has a volume of 60.

Base area
6
Base perimeter
12
Lateral area
120
Total surface area
132

Volume: 60. A triangular prism 10 long with a base area of 6 has a volume of 60.

How to calculate

Finds the volume, base area, lateral area and total surface area of a triangular prism from the three sides of its base, a base and height, or the legs of a right triangle, and its length.

Example with the default inputs (Triangle known by Three sides, Side a 3, Side b (base) 4, Side c 5, Prism length 10): A triangular prism 10 long with a base area of 6 has a volume of 60.

Method: V = B × L; SA = 2B + (a + b + c) × L. B = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2 (three sides), B = ½ × b × h (base and height), or B = ½ × a × b with c = √(a² + b²) (right triangle).

  • The prism is a right prism: the rectangular faces are at right angles to the two equal triangular ends.
  • All lengths are in the same unit; areas come out in that unit squared and the volume in that unit cubed.
  • With a base and height only, the other two sides are unknown, so the page gives the volume and base area but not the surface area.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Triangle known by Base and height, Side b (base) 12, Triangle height 6, Prism length 10 gives Base area 36, Volume 360.Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism: V = B × h, SA = 2B + ph), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area (Example 10.57: base area ½ × 12 × 6 = 36 in²)
  2. Triangle known by Three sides, Side a 10, Side b (base) 15, Side c 7, Prism length 20 gives Base area 29.393877, Volume 587.877538, Base perimeter 32, Lateral area 640.Source: OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (Heron’s formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines (s = 16, area = √(16 × 6 × 1 × 9) ≈ 29.4); OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism: V = B × h, SA = 2B + ph), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area
  3. Triangle known by Right triangle, Side a 3, Side b (base) 4, Prism length 10 gives Hypotenuse 5, Base area 6, Volume 60, Base perimeter 12, Lateral area 120, Total surface area 132.Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism: V = B × h, SA = 2B + ph), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area
  4. Triangle known by Three sides, Side a 2.5, Side b (base) 2.5, Side c 2.5, Prism length 4 gives Base area 2.706329, Volume 10.825318, Total surface area 35.412659.Source: OpenStax, Algebra and Trigonometry 2e, §10.2 Non-right Triangles: Law of Cosines (Heron’s formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines; OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism: V = B × h, SA = 2B + ph), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area

How it works

The prism has two equal triangular ends joined by three rectangles. Let B be the area of one triangle, a, b and c its sides, and L the length of the prism.

  • Volume V = B × L.
  • Lateral area = (a + b + c) × L, the three rectangles.
  • Total surface area SA = 2B + (a + b + c) × L.

The triangle area B comes from one of three modes:

  • Three sides: Heron’s formula, B = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2. The page works out 16B² = (a + b + c)(−a + b + c)(a − b + c)(a + b − c) exactly and takes one square root.
  • Base and height: B = ½ × b × h. The other two sides are unknown, so this mode gives only the volume and the base area.
  • Right triangle: legs a and b, B = ½ × a × b, and the hypotenuse c = √(a² + b²).

Rules. Every length is more than 0 and at most 1,000,000. In the three sides mode each side must be shorter than the other two added together; otherwise the sides make no triangle and the page gives no answer.

Exact arithmetic. Each length is read as the exact decimal you typed. Sums and products are exact fractions; a square root is exact when the number is a perfect square of a fraction (√25 = 5), and otherwise a double-precision float.

Output format. Decimals and significant figures below are the most shown; trailing zeros are dropped (23.0 shows as 23, money keeps its cents). Every result shows at most 4 decimals, rounded half up from its decimal value. Areas are in square units and the volume in cubic units of the length unit you used.

Assumptions

  • The prism is a right prism: the rectangles meet the triangular ends at right angles. An oblique prism with the same base and perpendicular length has the same volume, but a larger lateral area.
  • All lengths use the same unit.

Worked examples by hand

Base 12 and height 6, length 10 (OpenStax Example 10.57). B = ½ × 12 × 6 = 36. V = 36 × 10 = 360.

Sides 10, 15 and 7, length 20. s = 16; B = √(16 × 6 × 1 × 9) = √864 = 29.3939. V = 20 × √864 = 587.8775. Perimeter 32; lateral area 32 × 20 = 640.

A 3-4-5 right triangle, length 10. c = √(3² + 4²) = 5; B = ½ × 3 × 4 = 6; V = 60; perimeter 12; lateral area 120; SA = 2 × 6 + 120 = 132.

An equilateral triangle of side 2.5, length 4. s = 3.75; B = √(3.75 × 1.25³) = (√3 ÷ 4) × 6.25 = 2.7063; V = 10.8253; SA = 2 × 2.7063 + 7.5 × 4 = 35.4127.

Other questions people ask

How do I find the volume of a triangular prism?

Find the area of the triangle at one end, then multiply by the length of the prism. A triangle with base 12 in and height 6 in has an area of ½ × 12 × 6 = 36 in², so a prism 10 in long holds 36 × 10 = 360 in³.

How do I find the surface area of a triangular prism?

Add the two triangular ends and the three rectangles: SA = 2B + (a + b + c) × L. For a 3-4-5 right triangle and a length of 10, that is 2 × 6 + 12 × 10 = 132 square units.

What if I only know the three sides of the triangle?

Use Heron’s formula. Let s = (a + b + c) ÷ 2; then the area is √(s(s − a)(s − b)(s − c)). For sides 10, 15 and 7, s = 16 and the area is √864 ≈ 29.39.

Why is there no surface area in the base and height mode?

A base and a height fix the area of the triangle, but not the lengths of its other two sides, and the rectangles on those sides need them. Use the three sides mode to get the surface area.

What units does the calculator use?

Any unit you like, as long as every length uses the same one. The areas come out in that unit squared and the volume in that unit cubed: inches give in² and in³, centimetres give cm² and cm³.

What is the difference between the length and the height?

The triangle height is measured inside the triangle, at right angles to its base side. The prism length is the distance between the two triangular ends. Some books call the prism length its height.