Volume of a Pyramid Calculator
Calculate the volume, surface area, and slant height of a rectangular base pyramid from its base dimensions and height. See also Volume of Cone Calculator and Volume of Triangular Prism Calculator.
How to Calculate the Volume of a Pyramid
To find the volume of a rectangular base pyramid, multiply the base area (length × width) by the perpendicular height, then divide by 3. The pyramid volume is exactly one-third of a rectangular prism with the same base and height. For a square base pyramid, the length and width are equal. This calculator also computes the total surface area and slant heights.
Pyramid Volume Formula
V = (1/3) × l × w × h
Base Area = l × w
Slant Height (along l) = √(h² + (w/2)²)
Slant Height (along w) = √(h² + (l/2)²)
SA = l×w + l×sl + w×sw
(where sl and sw are the respective slant heights)
Example
Find the volume of a pyramid with base 6×6 and height 8:
V = (1/3) × l × w × h
V = (1/3) × 6 × 6 × 8
V = (1/3) × 288
V = 96 cubic units
Pyramid Volume Reference Table
| Base (L × W) | Height | Volume | Surface Area |
|---|---|---|---|
| 3 × 3 | 4 | 12.0000 | 34.6320 |
| 4 × 4 | 5 | 26.6667 | 59.0813 |
| 5 × 5 | 6 | 50.0000 | 90.0000 |
| 6 × 6 | 8 | 96.0000 | 138.5280 |
| 6 × 4 | 8 | 64.0000 | 107.6533 |
| 8 × 6 | 10 | 160.0000 | 196.1444 |
| 8 × 8 | 10 | 213.3333 | 236.3253 |
| 10 × 8 | 12 | 320.0000 | 310.4911 |
| 10 × 10 | 12 | 400.0000 | 360.0000 |
| 10 × 10 | 15 | 500.0000 | 416.2278 |
| 12 × 10 | 15 | 600.0000 | 471.2916 |
| 12 × 12 | 16 | 768.0000 | 554.1122 |
| 15 × 10 | 12 | 600.0000 | 486.5097 |
| 15 × 15 | 20 | 1500.0000 | 865.8003 |
| 20 × 20 | 25 | 3333.3333 | 1477.0330 |
Real-World Applications
Egyptian Pyramids
Archaeologists calculate the stone volume of ancient pyramids to estimate construction effort and workforce requirements.
Hip Roofs
Builders calculate the air space under pyramid-shaped roofs for insulation and ventilation planning.
Tent Peaks
Outdoor gear designers calculate internal headroom volume for pyramid-style tent designs.
Crystal Structures
Mineralogists calculate volumes of pyramid-shaped crystal formations for density measurements.
Solved Examples
Example 1: Great Pyramid of Giza
The Great Pyramid has an approximate square base of 230 m × 230 m and height of 146 m. Estimate its volume.
V = (1/3) × 230 × 230 × 146
V = (1/3) × 7,727,800
V ≈ 2,575,933 m³ (about 2.58 million cubic meters)
Example 2: Pyramid Roof Volume
A pyramid-shaped roof has a rectangular base of 8 m × 6 m and a peak height of 3 m. Find the air space volume.
V = (1/3) × 8 × 6 × 3
V = (1/3) × 144
V = 48 m³
Example 3: Glass Pyramid Display
A decorative glass pyramid has a square base of 12 cm × 12 cm and height of 20 cm. What is its volume?
V = (1/3) × 12 × 12 × 20
V = (1/3) × 2880
V = 960 cm³
Practice Questions
Q1: Find the volume of a pyramid with base 10 × 10 and height 12.
Answer: V = (1/3) × 10 × 10 × 12 = (1/3) × 1200 = 400 cubic units
Q2: A pyramid has volume 200 cm³, base length 10 cm, and base width 5 cm. What is the height?
Answer: h = 3V / (l×w) = 3×200 / (10×5) = 600/50 = 12 cm
Q3: A rectangular prism and pyramid have the same base (8×6) and height (9). Compare their volumes.
Answer: Prism V = 8×6×9 = 432; Pyramid V = 432/3 = 144. Pyramid is 1/3 the prism.
Q4: Find the slant height of a square pyramid with base side 10 and height 12.
Answer: Slant height = √(h² + (s/2)²) = √(144 + 25) = √169 = 13 units
Q5: A pyramid has a square base. If its volume is 972 cm³ and height is 9 cm, find the base side.
Answer: Base area = 3V/h = 3×972/9 = 324 cm²; side = √324 = 18 cm
Key Takeaways
- Pyramid volume = (1/3) × base area × height — always one-third of the equivalent prism.
- The 1/3 factor applies to ALL pyramids regardless of base shape (square, rectangular, triangular, etc.).
- Height must be perpendicular — measured from base to apex along a vertical line.
- Slant height is different from perpendicular height — it runs along a face.
- For square pyramids, set length = width in the formula.
- The Great Pyramid of Giza contains about 2.58 million m³ of stone.
Frequently Asked Questions
What is the volume of a pyramid?
The volume of a pyramid is the total three-dimensional space enclosed within its base and triangular faces meeting at the apex. For a rectangular base, V = (1/3) × length × width × height.
Why is the pyramid volume one-third of a prism?
This can be proven using calculus or by decomposing a rectangular prism into three pyramids of equal volume. The factor of 1/3 applies to all pyramids regardless of base shape.
What is the slant height?
The slant height is the distance from the apex to the midpoint of a base edge, measured along the triangular face. A rectangular pyramid has two different slant heights — one for each pair of opposite faces.
Does this work for a square pyramid?
Yes. A square pyramid is a special case where the base length and width are equal. Enter the same value for both to calculate a square pyramid.
What about triangular or hexagonal base pyramids?
The general formula V = (1/3) × base area × height works for any base shape. This calculator specifically handles rectangular bases. For other bases, calculate the base area separately and multiply by h/3.