About the Pyramid Calculator
This pyramid calculator works for any right pyramid with a square or rectangular base — the apex sits directly above the centre of the base. Enter the base length, base width and vertical height to get the volume, both slant heights, the length of the corner (lateral) edges, the lateral surface area and the total surface area.
It is useful for geometry classes, for estimating material in a pyramid-shaped roof, skylight, tent or planter, and for fun comparisons such as the volume of the Great Pyramid of Giza. For a square pyramid simply enter the same value for length and width.
Height here means the perpendicular height from the base to the apex. If you have measured the slant height up the middle of a face instead, the height is √(slant² − (half the base side)²).
Litre capacity uses the exact definitions 1 in = 2.54 cm (so 1 in³ = 16.387064 mL, 1 ft³ = 28.316846592 L, 1 yd³ = 764.554857984 L) and 1 US gallon = 3.785411784 L. With generic units the litre output is hidden because the size has no physical scale.
With the default inputs, the volume is 400. Change any value above to recalculate instantly.
How to use the pyramid calculator
- 1Enter the base length and width (use the same value for a square pyramid).
- 2Enter the perpendicular height from the base to the apex.
- 3Choose a real length unit (mm, cm, m, in, ft or yd) if you also want the capacity in litres.
- 4Read the volume, slant heights and surface areas.
Formula and method
Every pyramid has one third of the volume of the prism with the same base and height, so V = base area × height ÷ 3. For a rectangular base the base area is l × w.
Each triangular face has its own slant height — the height of that triangle measured up the middle of the face. The faces standing on the length sides rise over a horizontal run of w/2, and those on the width sides over l/2, so each slant height comes from the Pythagorean theorem. Two faces of area l·s_l/2 and two of area w·s_w/2 give the lateral area; adding the base gives the total surface area. The corner edge length is √(h² + (l/2)² + (w/2)²).
- l, w
- Base length and width
- h
- Perpendicular height to the apex
- s_l, s_w
- Slant heights of the faces on the length and width sides
- L
- Lateral surface area (four triangular faces)
- S
- Total surface area
Worked examples
Square pyramid, base 10, height 12
V = 10 × 10 × 12 / 3 = 400 cubic units. Each face rises 12 over a run of 5, so the slant height is √(144 + 25) = 13, and the four faces total 4 × (10 × 13 / 2) = 260. Adding the 100-unit base gives 360.
Great Pyramid of Giza (approx. 230 m base, 146.6 m original height)
Using the commonly quoted approximate original dimensions (the same 146.6 m figure as in the FAQ below; published estimates vary slightly), V = 230² × 146.6 / 3 ≈ 2.59 million m³ (about 2.59 billion litres). Each face has a slant height of about 186.3 m, the corner edges are about 219.0 m long, and the four faces cover roughly 85,700 m².
Rectangular pyramid 8 × 6, height 10
V = 8 × 6 × 10 / 3 = 160. The faces on the 8-unit sides have slant height √(100 + 9) ≈ 10.44, those on the 6-unit sides √(100 + 16) ≈ 10.77, so the lateral area is 8 × 10.44 + 6 × 10.77 ≈ 148.14.
Frequently asked questions
What is the formula for the volume of a pyramid?+
V = (base area × height) / 3. For a square base of side a it is a²h/3, and for a rectangular base l × w it is l·w·h/3 — one third of the matching box.
What is the difference between height and slant height?+
The height is measured straight down from the apex to the centre of the base. The slant height runs from the apex down the middle of a triangular face to the base edge, so it is always longer than the height.
How do you find the surface area of a square pyramid?+
Surface area = a² + 2a·s, where a is the base side and s is the slant height. For a = 10 and h = 12, s = 13, so the surface area is 100 + 260 = 360 square units.
Does this work for a triangular or hexagonal pyramid?+
This calculator covers square and rectangular bases. For any other base the volume is still base area × height / 3, so you can combine a regular-polygon area with that formula.
How big is the Great Pyramid of Giza?+
Its original base was about 230 m on each side and its height about 146.6 m, giving a volume of roughly 2.6 million cubic metres. Erosion and the lost capstone have reduced the present height to about 139 m.