Pyramid Volume Calculator
Calculate volume, surface area, slant height and lateral edge for square, rectangular, triangular and hexagonal pyramids — with full step-by-step working. Universal formula: V = (1/3) × base area × height for all pyramid types.
Square Pyramid — h = vertical height, a = base side, l = slant height
Universal Rule: V = (1/3) × Abase × h applies to ALL pyramids — square, rectangular, triangular, hexagonal, pentagonal, or any polygon base.
Real pyramid dimensions computed with V = (1/3) × base area × height. Click any row to load into the square pyramid calculator above.
| Pyramid | Base | Height | Volume |
|---|---|---|---|
| ⚡ Great Pyramid of Giza | 230.4m × 230.4m | 138.8m | ≈ 2.45 × 10⁶ m³ |
| Pyramid of Khafre | 215.5m × 215.5m | 136.4m | ≈ 2.11 × 10⁶ m³ |
| Pyramid of Menkaure | 108.5m × 108.5m | 65.5m | ≈ 2.57 × 10⁵ m³ |
| Louvre Pyramid (glass) | 35m × 35m | 21.6m | ≈ 8,820 m³ |
| Transamerica Pyramid | 53m × 53m | 260m | ≈ 2.43 × 10⁵ m³ |
| Pyramid of the Sun (Teotihuacan) | 213m × 213m | 65m | ≈ 9.84 × 10⁵ m³ |
⚡ Click highlighted rows to load into calculator. Volumes computed using V = (1/3)a²h.
Pyramid Volume Formula — The Universal Rule V = (1/3)Ah
The pyramid volume formula is one of the most elegant results in geometry: V = (1/3) × base area × height, written as V = (1/3)Ah. This single formula works for all pyramid types — square, rectangular, triangular, hexagonal, and any polygon base. The pyramid volume calculator above applies this universal rule to all four base types.
Why Is There a 1/3 Factor?
The 1/3 comes from a beautiful geometric fact: exactly three congruent pyramids can be assembled to fill a rectangular prism of the same base and height. Since volume of prism = A×h and three identical pyramids fill it, each pyramid has volume (1/3)×A×h. This dissection proof works for any base shape. The same 1/3 factor appears in the cone volume formula V = (1/3)πr²h because a cone is simply a pyramid with a circular base.
Deep connection: Cone formula V = (1/3)πr²h and pyramid formula V = (1/3)Ah are the SAME formula. A cone IS a pyramid with infinitely many sides on its base. Both arise from the same 1/3 dissection principle.
| Pyramid Type | Base Area A | Volume Formula |
|---|---|---|
| Square (side a) | a² | V = a²h/3 |
| Rectangular (l × w) | l × w | V = lwh/3 |
| Equilateral Triangle (side a) | (√3/4)a² | V = (√3/12)a²h |
| Hexagonal (side a) | (3√3/2)a² | V = (√3/2)a²h |
| Cone (radius r) | πr² | V = πr²h/3 |
Square Pyramid — Volume, Surface Area & Slant Height
The square pyramid is the most commonly studied — it has a square base with side length a, vertical height h, and four identical triangular faces. The pyramid volume calculator handles all square pyramid properties: volume, slant height, lateral edge, lateral area and total surface area.
Slant Height vs Lateral Edge — Critical Distinction
- Slant height (l): Distance from apex to the midpoint of a base edge. Runs along the surface. Used in area formulas.
l = √(h² + (a/2)²) - Lateral edge (e): Distance from apex to a base corner. Always longer.
e = √(h² + a²/2) - Common mistake: confusing these two — slant height is to edge midpoint, lateral edge is to corner.
Classic Example: a = 6m, h = 4m (3-4-5 Pythagorean Triple!)
- Base area: A = a² = 6² = 36 m²
- Slant height: l = √(4² + 3²) = √(16 + 9) = √25 = 5 m — a perfect 3-4-5 triple!
- Volume: V = (1/3) × 36 × 4 = 48 m³
- Lateral area: LA = 2al = 2 × 6 × 5 = 60 m²
- Total surface area: SA = a² + 2al = 36 + 60 = 96 m²
- Lateral edge: e = √(16 + 18) = √34 ≈ 5.831 m
How to Find the Volume of a Rectangular Pyramid
A rectangular pyramid has a rectangular base with length l and width w. The pyramid volume formula becomes V = (1/3)lwh. A key difference from square pyramids: rectangular pyramids have two different slant heights — one for each pair of opposite faces.
Worked Example: l = 8m, w = 6m, h = 10m
- Base area: A = l × w = 8 × 6 = 48 m²
- Volume: V = (1/3) × 48 × 10 = 160 m³
- Slant height 1 (over length-faces): l₁ = √(10² + (6/2)²) = √(100 + 9) = √109 ≈ 10.44 m
- Slant height 2 (over width-faces): l₂ = √(10² + (8/2)²) = √(100 + 16) = √116 ≈ 10.77 m
- Lateral area: LA = l×l₁ + w×l₂ = 8×10.44 + 6×10.77 = 83.52 + 64.62 ≈ 148.14 m²
- Total SA: SA = lw + LA = 48 + 148.14 ≈ 196.14 m²
Pyramid Surface Area — Lateral Area and Total Surface Area
The pyramid surface area has two components: the base area and the lateral surface area (the sum of all triangular faces). Understanding slant height is essential — it is the height of each triangular face.
Square Pyramid Surface Area Formulas
- Each triangular face area: (1/2) × base × slant height = (1/2) × a × l
- Lateral area (4 faces): LA = 4 × (1/2)al = 2al
- Total SA: SA = a² + 2al (base + lateral)
⚠️ Most common mistake: Using the vertical height h instead of the slant height l when calculating face areas. The triangular faces are slanted — their height measured along the face is l (slant height), NOT h (vertical height). Always compute l = √(h² + (a/2)²) first.
Why Slant Height, Not Vertical Height?
- Each triangular face is a triangle with base = a (base edge) and height = l (slant height)
- The slant height l goes from the apex down to the midpoint of the base edge — along the surface
- The vertical height h is perpendicular to the base — it goes straight down inside the pyramid
- Using h instead of l underestimates the lateral area because h < l always
The Great Pyramid of Giza — A Worked Example
The Great Pyramid of Giza is the world's most famous square pyramid — and the pyramid volume formula gives us its exact volume. Base side a = 230.4m, height h = 138.8m (original). This makes the pyramid volume calculator tangible and memorable.
Great Pyramid Volume Calculation
- Base area: A = a² = 230.4² = 53,084.16 m²
- Volume: V = (1/3) × 53,084.16 × 138.8 = (1/3) × 7,368,481.0 ≈ 2,456,160 m³ ≈ 2.45 × 10⁶ m³
- Slant height: l = √(138.8² + (230.4/2)²) = √(19,265.44 + 13,271.04) = √32,536.48 ≈ 180.4 m
- Lateral area: LA = 2 × 230.4 × 180.4 ≈ 83,160 m²
- Total SA: SA = 53,084 + 83,160 ≈ 136,244 m² (13.6 hectares of stone!)
Pyramid vs Cone — The 1/3 Connection
The pyramid volume formula V = (1/3)Ah and the cone formula V = (1/3)πr²h share the same 1/3 factor — and this is no coincidence. A cone is simply a pyramid whose base is a circle (A = πr²) rather than a polygon. Both shapes follow the universal rule: volume = (1/3) × base area × height.
| Shape | Base Area | Volume Formula | Same as V = (1/3)Ah? |
|---|---|---|---|
| Square pyramid | a² | (1/3)a²h | ✓ Yes |
| Rectangular pyramid | lw | (1/3)lwh | ✓ Yes |
| Triangular pyramid | (√3/4)a² | (√3/12)a²h | ✓ Yes |
| Cone | πr² | (1/3)πr²h | ✓ Yes (circle base) |
| Prism / Cylinder | Varies | Ah (no 1/3) | ✗ No 1/3 |
Memory trick: Prisms and cylinders have volume = A × h (no 1/3). Pyramids and cones have volume = (1/3) × A × h. The pointy shape loses 2/3 of the volume compared to a prism of the same base and height.
Common Mistakes in Pyramid Calculations
❌ Mistake 1 — Using Vertical Height h Instead of Slant Height l in Lateral Area
- Wrong: LA = 2 × a × h (uses vertical height h)
- Correct: LA = 2 × a × l, where l = √(h² + (a/2)²) (slant height)
- For a=6, h=4: wrong gives LA = 48 m², correct gives LA = 2×6×5 = 60 m²
❌ Mistake 2 — Forgetting the 1/3 Factor in Volume
- Wrong: V = a²h (this is the prism volume, 3× too large)
- Correct: V = (1/3)a²h
- For a=6, h=4: wrong gives V = 144 m³, correct gives V = 48 m³
❌ Mistake 3 — Using One Slant Height for Both Face Pairs in Rectangular Pyramids
- Wrong: assuming both pairs of faces have the same slant height
- Correct: l₁ = √(h² + (w/2)²) for length-faces, l₂ = √(h² + (l/2)²) for width-faces
- These are only equal when l = w (square pyramid)
❌ Mistake 4 — Forgetting to Add the Base in Total Surface Area
- Wrong: SA = 2al (lateral area only, missing base)
- Correct: SA = a² + 2al (base area + lateral area)
- For a=6, l=5: wrong gives SA = 60 m², correct gives SA = 96 m²
❌ Mistake 5 — Not Rewriting 1/x² Before Applying Area Formula
- Wrong: confusing base perimeter with base area in volume formula
- Correct: volume needs base AREA (a² for square, lw for rectangle), not perimeter (4a or 2l+2w)
- The formula is V = (1/3) × AREA × height, not V = (1/3) × perimeter × height