Triangular Pyramid Volume Calculator

Four triangular faces. One apex. One formula. The triangular pyramid volume calculator handles every possible input scenario — whether you know the base area already or need to calculate it from your triangle's dimensions using four different triangle methods.

What Is a Triangular Pyramid?

A triangular pyramid is a three-dimensional solid with:

• A triangular base

• Three triangular faces meeting at a single point (apex)

• Also called a tetrahedron when all four faces are equilateral triangles

Its volume is exactly one-third of a triangular prism with the same base and height — the same relationship that exists between all pyramids and their corresponding prisms.

Formula for Volume of a Triangular Pyramid

V = ⅓ × A × h

Where:

• V = Volume (cubic units — cm³, m³, in³)

• A = Area of the triangular base

• h = Perpendicular height of the pyramid (base to apex)

Base Area Formula

A = ½ × b × h_triangle

Where:

• b = base of the triangle

• h_triangle = height of the triangular base

So the complete combined formula is:

V = ⅓ × (½ × b × h_triangle) × H

Or simplified:

V = (b × h_triangle × H) ÷ 6

Where H = pyramid height.

How the Triangular Pyramid Volume Calculator Works

Step 1 — Do You Know the Base Area?

Option A — "I know the area of the pyramid's base" Enter the base area directly — skip triangle calculation entirely.

Option B — "I don't know the area of the pyramid's base" (default) Select your triangle type and enter the relevant dimensions.

Step 2 — Select Triangle Type

Step 3 — Enter Pyramid Height

• Pyramid base area — auto-calculated from triangle inputs

• Pyramid height (H) — enter the perpendicular height of the pyramid

Output

• Volume — displayed in cm³ (auto-calculated)

How to Calculate Triangular Pyramid Volume — Step by Step

Example 1 — Using Calculator Default Values

Base (b) = 10 cm, Base height (h) = 20 cm, Pyramid height (H) = 22 cm

Step 1 — Calculate base area: A = ½ × b × h A = ½ × 10 × 20 A = 100 cm² ✅ Matches calculator output exactly

Step 2 — Calculate volume: V = ⅓ × A × H V = ⅓ × 100 × 22 V = ⅓ × 2,200 V = 733.33 cm³ ✅ Matches calculator output exactly

Example 2 — Smaller Pyramid

Base = 6 cm, Base height = 8 cm, Pyramid height = 15 cm

1. A = ½ × 6 × 8 = 24 cm²

2. V = ⅓ × 24 × 15

3. V = ⅓ × 360

4. V = 120 cm³

Example 3 — If Base Area Already Known

Base area = 50 cm², Pyramid height = 12 cm

1. V = ⅓ × 50 × 12

2. V = ⅓ × 600

3. V = 200 cm³

Triangular Pyramid Volume — Quick Reference Table

All values calculated using V = (b × h × H) ÷ 6

Why Is There a 1/3 in the Volume Formula for All Pyramids?

The factor of ⅓ appears because any pyramid — triangular, square, or otherwise — occupies exactly one-third of the space of a prism with the same base and height.

This can be demonstrated physically: it takes exactly three identical pyramid-shaped containers to fill one prism-shaped container of the same base and height.

Mathematically, the ⅓ comes from integral calculus — summing infinitely thin cross-sectional slices from base to apex, where each slice's area decreases as a square function of height, integrating to exactly one-third of the full prism volume.

The same ⅓ rule applies to:

• Triangular pyramids ✅

• Square pyramids ✅

• Rectangular pyramids ✅

• Cones ✅ (which are circular pyramids)

Fun Fact That'll Make You Laugh 😄

The Great Pyramid of Giza — the most famous pyramid in the world — is a square pyramid, not a triangular one.

But its volume is still calculated using the same ⅓ rule: V = ⅓ × base area × height.

Ancient Egyptians built a structure containing approximately 2.6 million cubic meters of stone using this relationship — without calculators, without algebra, and without anyone explaining why the ⅓ works.

They just knew it did. 😂

Frequently Asked Questions

Why is there a 1/3 in the volume formula for all pyramids?

Because a pyramid occupies exactly one-third of the volume of a prism with the same base and height. Three identical pyramids fit perfectly inside one equivalent prism — demonstrated physically or proven mathematically through calculus integration of decreasing cross-sectional areas from base to apex.

What is the formula for triangular pyramid volume?

V = ⅓ × A × h, where A is the triangular base area and h is the pyramid's perpendicular height. If the base area isn't known: V = (b × h_triangle × H) ÷ 6, where b and h_triangle define the base triangle and H is the pyramid height.

What is the difference between a triangular pyramid and a tetrahedron?

A triangular pyramid has a triangular base and three triangular faces — any triangle can form the base. A regular tetrahedron is a special case where all four faces are equilateral triangles of equal size. Every tetrahedron is a triangular pyramid, but not every triangular pyramid is a tetrahedron.

How does the SSS triangle method work in the calculator?

SSS (Side-Side-Side) uses all three side lengths of the triangular base to calculate its area using Heron's formula — no angles needed. The calculator handles this automatically when you select the SSS option and enter the three side lengths.

What units does the triangular pyramid volume calculator use?

The calculator uses centimeters (cm) for length inputs and outputs volume in cubic centimeters (cm³). You can apply the same formula using any consistent unit — meters, inches, or feet — and the output will be in the corresponding cubic unit.