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Triangular prism volume, cylinders, cones and other 3D shapes

How to find the volume and surface area of a triangular prism, and the volume of cylinders and cones — one idea (area of the cross-section × length) behind most of them.

1 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

The volume of any prism is the area of its cross-section times its length. For a triangular prism volume, that means: find the area of the triangle, then multiply by the length of the prism. Cylinders work the same way with a circle; cones and pyramids take one third of that.

Vprism=Across-section×L=12bh×LV_{\text{prism}} = A_{\text{cross-section}} \times L = \tfrac{1}{2} b h \times L

Worked example

Easy: triangular prism volume

A triangular prism has a triangle with base 6 cm and height 4 cm, and length 10 cm. Find its volume.

Show the working
  1. Area of the triangle.
    12×6×4=12 cm2\tfrac{1}{2} \times 6 \times 4 = 12\text{ cm}^2
  2. Multiply by the length.
    12×10=120 cm312 \times 10 = 120\text{ cm}^3

Answer: 120 cm3120\text{ cm}^3

Checked by Sumzo’s math engine

Surface area of a right triangular prism

A right triangular prism has two triangle ends and three rectangular faces. Add the two triangles to the three rectangles; the rectangles together are the perimeter of the triangle times the length.

Worked example

Medium: surface area

A right triangular prism has a 3–4–5 right triangle as its end and length 10 cm. Find its surface area.

Show the working
  1. Two triangles.
    2×12×3×4=122 \times \tfrac{1}{2} \times 3 \times 4 = 12
  2. Three rectangles: the perimeter of the triangle times the length.
    (3+4+5)×10=120(3 + 4 + 5) \times 10 = 120
  3. Add.
    12+120=132 cm212 + 120 = 132\text{ cm}^2

Answer: 132 cm2132\text{ cm}^2

Checked by Sumzo’s math engine

Cylinders

Worked example

Medium: volume of a cylinder

Find the volume of a cylinder with radius 2 cm and height 7 cm.

Show the working
  1. The cross-section is a circle, area πr2\pi r^2.
    V=πr2h=π×4×7V = \pi r^2 h = \pi \times 4 \times 7
  2. Simplify.
    28π≈87.96 cm328\pi \approx 87.96\text{ cm}^3

Answer: 28π≈87.96 cm328\pi \approx 87.96\text{ cm}^3

Checked by Sumzo’s math engine

Volume of a circular cone

Worked example

Hard: a cone

Find the volume of a circular cone with radius 3 cm and height 4 cm.

Show the working
  1. A cone is one third of the cylinder with the same base and height.
    V=13πr2h=13π×9×4V = \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3}\pi \times 9 \times 4
  2. Simplify.
    12π≈37.70 cm312\pi \approx 37.70\text{ cm}^3

Answer: 12π≈37.70 cm312\pi \approx 37.70\text{ cm}^3

Checked by Sumzo’s math engine

Common mistakes

  • Forgetting the 12\frac{1}{2} in the triangle’s area.
  • Using the slant height of a cone instead of its perpendicular height.
  • Counting only one triangular end when finding surface area.
  • Mixing cm² (area) and cm³ (volume).

Frequently asked questions

What is the formula for triangular prism volume?

V=12×b×h×LV = \frac{1}{2} \times b \times h \times L: the area of the triangular end times the length of the prism.

What is a right triangular prism?

A prism whose rectangular faces meet the triangular ends at right angles. People also use the name for a prism whose triangle has a right angle.

How do I find the volume of a cone?

V=13πr2hV = \frac{1}{3}\pi r^2 h, one third of the cylinder with the same radius and height.

Calculators

Practice

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