Pyramid Volume Formula: Calculate Any Pyramid in 3D
Explore the universal formula $$V = \frac{1}{3} A_b h$$ across all pyramid geometries: Square, Rectangular, Triangular (Tetrahedron), Hexagonal, and Truncated Frustums. Visualize real-time 3D dimensions, manipulate models, examine 2D cross-sections, and see the interactive water pouring proof.
Interactive Pyramid Volume Calculator
Choose your pyramid base geometry, enter parameters with synchronized sliders, and get instant volume, surface areas, slant heights, and KaTeX step-by-step arithmetic.
Square Pyramid Volume
Base side length $a$ and vertical perpendicular height $h$.
Step-by-Step Mathematical Solution
Detailed mathematical substitution and algebraic steps for your inputs.
Equivalent Volume in Other Units
Explore All 16 Dedicated Pyramid Calculators
Need a solver for a triangular base, regular hexagon, truncated frustum, or hip roof? Choose from our complete suite of 16 specialized, step-by-step pyramid calculators.
Pyramid Volume Calculator
Triangular Pyramid Volume
Square Pyramid Volume
Rectangular Pyramid
Pentagonal Pyramid
Hexagonal Pyramid
Frustum of a Pyramid
Irregular Pyramid
Oblique Pyramid
Polygonal Pyramid
Pyramid Surface Area
Half Pyramid Volume
Regular Tetrahedron
Inverted Pyramid Volume
Pyramid Slant Height
Volume by Base Shape
Why is the Volume of a Pyramid Exactly $\frac{1}{3}$ of a Prism?
Experience the physical water pouring experiment discovered by ancient Greek mathematicians and verified by calculus integration.
Water Pouring Experiment Simulation
Click the button below to pour water from 3 identical square pyramids into a square prism (box) of identical base and height ($a$ and $h$ equal).
Two Mathematical Verifications
Consider a cube with side length $s$. Connect the center of the cube to all 8 vertices.
- This divides the cube into 6 identical square pyramids, each with a base of area $s^2$ and height $h = s/2$.
- The volume of each of these 6 pyramids is: $$V_1 = \frac{\text{Volume of Cube}}{6} = \frac{s^3}{6} = \frac{s^2 \cdot (2h)}{6} = \frac{1}{3} s^2 h$$
- Similarly, a triangular prism can be cleanly cut into exactly 3 triangular pyramids of equal volume by drawing diagonals across its rectangular faces.
- Therefore: $$\text{Volume of Pyramid} = \frac{1}{3} \times \text{Base Area} \times h$$
Pyramid Formulas Matrix & Cheat Sheet
Formulas for every pyramid type, slant heights, lateral surface areas, total surface areas, and frustums.
Base is a square with side length $a$. Height $h$ is the vertical perpendicular distance from the center to the apex.
Base is a rectangle of length $l$ and width $w$. Features two distinct slant heights along each axis.
For a regular tetrahedron with 4 congruent equilateral triangle faces of edge $a$. For general triangular pyramid: $V = \frac{1}{3} A_b h$.
Base is a regular hexagon of side $a$ ($A_b = \frac{3\sqrt{3}}{2} a^2$). Multiplying by $\frac{1}{3} h$ yields the formula.
Top sliced parallel to base (hoppers, step pyramids). For square frustum: $V = \frac{1}{3} h (a_1^2 + a_2^2 + a_1 a_2)$.
Hypotenuse formed by height $h$ and base apothem $r_{in}$ (for square: $r_{in} = a/2$). Lateral Area = $\frac{1}{2} \cdot \text{Perimeter} \cdot s$.
Anatomy of a Pyramid & 2D Unfolded Net
Understand every edge, face, angle, and the geometric relationships inside a 3D pyramid.
Key Components of Every Pyramid
| Component | Symbol | Definition & Role |
|---|---|---|
| Apex (Vertex) | Point | Top-most point where all lateral triangular faces meet. |
| Base ($A_b$) | $a \times b$ | The flat polygon on bottom (square, rectangle, triangle, hexagon). |
| Perpendicular Height ($h$) | $h$ | Vertical distance from base center to apex at a 90° right angle. |
| Slant Height ($s$) | $s$ | Perpendicular altitude of a lateral triangular face: $s = \sqrt{h^2 + (a/2)^2}$. |
| Lateral Edge / Hip ($e$) | $e$ | Corner line segment connecting apex to a base corner vertex: $e = \sqrt{s^2 + (a/2)^2}$. |
| Lateral Face Area ($A_L$) | $A_L$ | Sum of triangular face areas: $n \times \frac{1}{2} a s = \frac{1}{2} P s$. |
Unfolded 2D Surface Net
A square pyramid unfolds into 1 central square base and 4 isosceles triangular wings.
Everything You Need to Know About Pyramids
Authoritative answers on calculations, slant heights, oblique solids, and historical applications.
The universal formula is $V = \frac{1}{3} A_b h$, where $A_b$ is the area of the base polygon and $h$ is the vertical perpendicular height from the center of the base to the apex. This rule applies regardless of whether the base is a square, rectangle, triangle, hexagon, or any polygon.
The vertical perpendicular height ($h$) is measured straight down inside the solid from the apex to the center of the base at a 90° right angle. The slant height ($s$) is measured along the exterior sloped face from the apex to the midpoint of a base edge. In a square pyramid, they form a right triangle: $s^2 = h^2 + (a/2)^2$. Use height for volume, and slant height for surface area!
Yes! By Cavalieri's Principle, two solids with equal base areas and equal heights have identical volumes if cross-sectional slices parallel to the base have equal areas at all heights. Even if the apex is shifted off to one side (an oblique pyramid), as long as $h$ is the true vertical perpendicular height, $V = \frac{1}{3} A_b h$ remains 100% exact.
A frustum is the solid remaining after the top apex of a pyramid is sliced off parallel to the base. Its volume is given by the formula $V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2})$, where $A_1$ is the lower base area, $A_2$ is the upper base area, and $h$ is the vertical height between the two parallel planes. This is widely used for roof hoppers and ancient step pyramids like the Mayan El Castillo.
A regular tetrahedron is a triangular pyramid where all four faces are congruent equilateral triangles with edge length $a$. Its volume formula simplifies directly to $V = \frac{a^3}{6\sqrt{2}} \approx 0.11785 \, a^3$. Its height is $h = a \sqrt{2/3} \approx 0.8165 a$.
Rearrange the volume formula by multiplying both sides by 3 and dividing by the base area $A_b$: $h = \frac{3V}{A_b}$. For a square pyramid with edge $a$, this becomes $h = \frac{3V}{a^2}$. You can use our "Solve Height from $V$" tab in the calculator above to calculate this automatically!
Lateral Area ($A_L$) is the surface area of only the side triangular faces (excluding the bottom base). For a regular pyramid, $A_L = \frac{1}{2} P s$, where $P$ is base perimeter and $s$ is slant height. Total Surface Area ($A_T$) adds the bottom base area: $A_T = A_b + A_L$. For a square pyramid, $A_T = a^2 + 2as$.