Volume of Triangular Pyramid
Updated on July 29, 2026
The volume of a triangular pyramid is found by multiplying the base area by one-third the height: V = (1/3) × B × h. This article covers the parts of a triangular pyramid, the full formula, worked examples, and practice problems.
What Is the Volume of a Triangular Pyramid?
The volume of a triangular pyramid is the amount of space it encloses — think of it as how much water the pyramid could hold if you filled it. Volume is always measured in cubic units, like cm³ or m³. A triangular pyramid holds exactly one-third the volume of a prism that shares the same triangular base and the same height — this one-third relationship is the key idea behind the formula.
A triangular pyramid is one member of a much larger family of 3D shapes, and the base-times-height logic used here shows up again with cubes, prisms, and cones.
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Triangular Pyramid and Its Parts
A triangular pyramid is a polyhedron made up of a triangular base and three triangular sides meeting at a single point. Its parts are:
- Base — the bottom face, always a triangle (equilateral, isosceles, or scalene)
- Apex — the topmost point where the three lateral faces meet
- Lateral faces — the three triangles connecting the base to the apex
- Height (h) — the perpendicular distance from the apex straight down to the base, forming a 90° angle with it
- Slant height (l) — the distance from the apex down the center of one lateral face (not the same as the height)
- Vertices and edges — 4 vertices (corners) and 6 edges in total
Mixing up slant height and true height is the most common mistake students make when calculating volume, so it’s worth double-checking which measurement a problem is giving you.
If your child is still getting comfortable naming the parts of solid figures in general, our shapes for kids guide is a good warm-up before tackling pyramid vocabulary specifically.
Volume of a Triangular Pyramid Formula
The formula is V = (1/3) × B × h, where V is volume, B is the area of the triangular base, and h is the pyramid’s vertical height. Once you know the base area and height, you can plug them straight into the formula like a calculator.
How to Find the Volume of a Triangular Pyramid
- Find the base area. Since the base is a triangle, multiply its base by its height and divide by 2.
- Identify the pyramid’s height — the vertical distance from the apex straight down to the base.
- Multiply the base area (B) by the height (h), then divide the result by 3.
For example, if the base area is 30 cm² and the pyramid is 10 cm tall: 30 × 10 = 300 cm³, then 300 ÷ 3 = 100 cm³. Always express the final answer in cubic units, since volume measures three-dimensional space.
Triangular Pyramid vs. Triangular Prism Volume
Since a triangular pyramid always holds one-third the volume of a prism with the same base and height, it helps to see the two formulas side by side.
| Shape | Volume Formula | Same B and h → Volume |
| Triangular prism | V = B × h | 300 cm³ (example) |
| Triangular pyramid | V = (1/3) × B × h | 100 cm³ (example) |
For the full breakdown of the prism side of this comparison, see our volume of prism guide.
Facts About the Volume of a Triangular Pyramid
- A triangular pyramid always occupies exactly one-third the volume of a triangular prism sharing the same base and height — it takes three full pyramids to fill an equivalent prism.
- Volume depends only on base area and perpendicular height. Whether the pyramid stands straight or leans, the volume stays the same as long as the base area and vertical height don’t change — a result known as Cavalieri’s Principle.
- When the base and all three lateral faces are identical triangles, the shape is called a regular tetrahedron. Any of its four faces can serve as the base, and the volume calculation always gives the same result.
- Because their volume is concentrated toward a wide base, triangular pyramids are structurally stable, which is part of why the shape appears often in engineering and architecture.
Students who want extra one-on-one practice with base-area and height calculations like these tend to benefit from a geometry tutor, especially once the topic moves from triangular pyramids into more complex solids.
Solved Examples for the Volume of a Triangular Pyramid
Solved example 1
A triangular pyramid has a base area of 24 cm² and a vertical height of 9 cm. Find its volume.
Solution: V = (1/3)Bh = (1/3)(24 × 9) = (1/3)(216) = 72 cm³.
Solved example 2
A triangular base has a base length of 10 cm and an altitude of 6 cm. The pyramid itself is 12 cm tall. Find the volume.
Solution: First, find the base area: B = (1/2)(10 × 6) = 30 cm². Then V = (1/3)Bh = (1/3)(30 × 12) = 120 cm³.
Solved example 3
A triangular pyramid has a volume of 90 cm³ and a base area of 15 cm². Find the height.
Solution: 90 = (1/3)(15 × h) → 270 = 15h → h = 18 cm.
Practice Problems on the Volume of a Triangular Pyramid
- A triangular pyramid has a base area of 15 cm² and a height of 6 cm. Find the volume.
- A triangular base has a base length of 8 in and an altitude of 3 in; the pyramid is 10 in tall. Find the volume.
- A regular tetrahedron has a base area of 21 m² and a height of 7 m. Find the volume.
Volume of Triangular Pyramid Worksheets
Put this into practice with related Brighterly worksheets:
- Surface Area and Volume Worksheets
- 7th Grade Volume and Surface Area Worksheets
- 3D Shapes Worksheet
- 2D and 3D Shapes Worksheets
Frequently Asked Questions on the Volume of a Triangular Pyramid
How Do You Find the Volume of a Triangular Pyramid?
Calculate the area of the triangular base first, then multiply that base area by the pyramid’s height (the vertical distance from apex to base) and divide by 3. Make sure to use the vertical height rather than the slant height — mixing the two up is the most common error.
What Is the Difference Between Surface Area and Volume of a Triangular Pyramid?
Volume measures the internal space of the pyramid — how much it could hold if filled with water. Surface area measures the total area covering its exterior, meaning the combined area of the base and all three lateral faces. The two use different formulas and different units (cubic vs. square).
What Is the Formula for the Volume of a Triangular Pyramid?
The formula is V = (1/3)Bh, where V is volume, B is the base area, and h is the height. Multiply the base area by the height, then divide the result by 3 to get the volume, expressed in cubic units.
Why Is the Volume of a Pyramid One-Third of a Prism’s?
A triangular pyramid and a triangular prism sharing the same base and height relate by a fixed 1:3 ratio — three pyramids of that size fit exactly inside one matching prism. This relationship holds for any pyramid and its matching prism, not just triangular ones, and can be shown using calculus or a geometric dissection argument.
Does a Triangular Pyramid Have the Same Volume Formula as a Rectangular Pyramid?
Both share the same one-third rule — V = (1/3)Bh — but the base area (B) is calculated differently for each. A triangular base uses (1/2) × base × height, while a rectangular base uses length × width. Once B is found correctly, the rest of the formula works the same way.
What Units Should You Use for Triangular Pyramid Volume?
Always use cubic units — cm³, m³, in³, or ft³ — since volume measures three-dimensional space. If the base and height are given in different units, convert them to match before applying the formula, since mismatched units will produce an incorrect result.
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