Geometry
How To Calculate The Volume Of A Pyramid
A pyramid holds exactly one third of the prism built on the same base with the same height, whatever the shape of that base.
Quick Answer
volume = base area x height / 3
- V
- Volume inside the pyramid
- B
- Area of the base
- h
- Perpendicular height from base to apex
Work out the base area, multiply by the height and divide by three. A pyramid on a six by four base with a height of nine holds 72 cubic units, exactly a third of the box it would fit inside.
What Is The Volume Of A Pyramid?
The volume of a pyramid is one third of the base area multiplied by the height. A pyramid on a base six by four with a height of nine holds seventy two cubic units, which is exactly a third of the box that contains it.
The one third is the same factor that turns a cylinder into a cone, and it is not a coincidence. Any shape whose cross section shrinks linearly to a point takes exactly one third of the volume of the prism around it, whether the base is a circle or a rectangle.
The base area is whatever the base happens to be. A square pyramid uses the side squared, a rectangular one uses length times width, and a triangular one uses half the base times the base height. The volume formula itself never changes.
The height is the perpendicular distance from the base to the apex, not the distance along a sloping edge. In a tall thin pyramid the slant edge is much longer than the height, and substituting it inflates the volume badly.
Surface area is a separate calculation and involves the slant heights of each triangular face. For a rectangular pyramid there are two different slant heights, one measured across the length and one across the width, which is why the surface area is fiddlier than the volume.
Each slant height comes from Pythagoras. It is the square root of the height squared plus half the relevant base dimension squared. For a base six by four with height nine, the slant along the width is about 9.2195 and along the length about 9.4868.
Pyramids scale dramatically with size. Doubling every dimension multiplies the volume by eight but the surface area by only four, which is why the largest pyramids in the world are enormous by any human standard and why they needed so much stone.
Real pyramid problems are usually about material or capacity, and those need different answers. The stone in the Great Pyramid is its volume, the cladding is its surface area, and a hopper shaped like an inverted pyramid holds its volume of grain.
Units work the usual way. Base dimensions and height in metres give a volume in cubic metres. Mixed units must be converted first, and the base area is a square measure while the volume is a cubic one.
A frustum, which is a pyramid with the top cut off, has its own formula involving both the top and bottom areas. Using the plain pyramid formula on a frustum badly underestimates the volume, and the error grows as the cut gets closer to the base.
Finally, note that the apex need not be above the centre of the base. A slanted or oblique pyramid with the same base and the same perpendicular height has exactly the same volume, because the volume depends on the height rather than on where the apex sits horizontally.
Formula
V = base area x height / 3
The prism formula divided by three. Works for any base shape, and for a slanted apex too.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| V | Volume | cubic metres | Space enclosed by the pyramid. |
| B | Base area | square metres | Area of the base, whatever shape it is. |
| h | Height | metres | Perpendicular distance from base to apex. |
l = sqrt((half base)^2 + h^2)
Pythagoras on the triangle from the base centre, up the height, and out to the midpoint of an edge.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| l | Slant height | metres | Distance up a triangular face from edge midpoint to apex. |
| b | Base dimension | metres | The base side measured across to that face. |
| h | Height | metres | Perpendicular height of the pyramid. |
A = base area + sum of triangular faces
Each face is a triangle of base b and slant height l. A rectangular pyramid has two different slant heights.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Surface area | square metres | Total outside area including the base. |
| B | Base area | square metres | Area of the base. |
| l | Slant height | metres | Slant height for the face in question. |
How To Calculate The Volume Of A Pyramid
- 1
Work out the area of the base
Length times width for a rectangle, side squared for a square, half base times height for a triangle. The volume formula accepts any of these.
- 2
Measure the perpendicular height
Straight up from the base to the apex, not along a sloping edge. If you only have a slant height, recover the perpendicular height with Pythagoras.
- 3
Multiply base area by height
Twenty four times nine is two hundred and sixteen. This is the volume of the prism, which is three times too large.
- 4
Divide by three
Two hundred and sixteen divided by three is seventy two cubic units. The division is what turns the prism into the pyramid that fits inside it.
- 5
Convert to litres if the answer is a capacity
Base dimensions in decimetres give litres directly, or convert cubic metres by multiplying by a thousand. The one third applies before any unit conversion, not after.
Examples
Example 1: A pyramid on a six by four base with height nine
- Base length
- 6
- Base width
- 4
- Height
- 9
| Step | Calculation | Result |
|---|---|---|
| Base area | 6 x 4 | 24 |
| Slant height across the width | sqrt(2^2 + 9^2) | 9.2195 |
| Volume | 24 x 9 / 3 | 72 |
Result: The pyramid holds 72 cubic units, exactly one third of the box six by four by nine that would enclose it.
Example 2: A square pyramid ten by ten with height twelve
- Base length
- 10
- Base width
- 10
- Height
- 12
| Step | Calculation | Result |
|---|---|---|
| Base area | 10 x 10 | 100 |
| Slant height | sqrt(5^2 + 12^2) | 13 |
| Volume | 100 x 12 / 3 | 400 |
Result: A square pyramid ten by ten with height twelve holds 400 cubic units, and its slant height is exactly 13.
Example 3: A small display pyramid five by three with height eight
- Base length
- 5
- Base width
- 3
- Height
- 8
| Step | Calculation | Result |
|---|---|---|
| Base area | 5 x 3 | 15 |
| Slant height across the width | sqrt(1.5^2 + 8^2) | 8.1394 |
| Volume | 15 x 8 / 3 | 40 |
Result: The display pyramid holds 40 cubic units, which is 40 litres if the measurements were in decimetres.
Calculator
Volume
72
- Base area
- 24
- Area of the four faces
- 93.2646
- Total surface area
- 117.2646
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Volume Of A Pyramid calculator page.
Common Mistakes
Forgetting the division by three
Without it you have the volume of the enclosing prism, which is exactly three times too large. It is the single most common error with this formula.
Using a slant height as the perpendicular height
The slant height runs up a face and is always longer than the vertical height. Substituting it overstates the volume, sometimes by a wide margin on a shallow pyramid.
Using the base perimeter instead of the base area
The formula needs the area. A perimeter is a length, and multiplying it by the height gives something with the wrong dimensions entirely.
Treating the apex position as important
A slanted pyramid with the same base and perpendicular height has the same volume. The horizontal position of the apex makes no difference at all.
Applying the formula to a frustum
A pyramid with the top cut off needs its own formula involving both the top and bottom areas. Using the plain pyramid formula underestimates it.
Assuming all four faces have the same slant height
A rectangular pyramid has two different slant heights, one across the length and one across the width. Using one for all four faces gives the wrong surface area.
Mixing units between the base and the height
A base in metres with a height in centimetres produces a meaningless result. Convert to a single unit before multiplying.
FAQ
What is the formula for the volume of a pyramid?
One third of the base area multiplied by the perpendicular height. It works for any base shape, whether square, rectangular or triangular.
Why is there a one third in the formula?
Because a pyramid occupies exactly one third of the prism with the same base and height. Three identical pyramids would fill that prism completely.
How do you find the height if you only know a slant height?
Use Pythagoras. The slant height, the perpendicular height and half the base dimension form a right triangle, so the height is the square root of the slant height squared minus half the base squared.
Does the apex have to be above the centre?
No. A slanted pyramid with the same base and the same perpendicular height has exactly the same volume, because only the height matters.
How do you find the surface area of a pyramid?
Add the base area to the areas of the triangular faces. Each face is half its base times the relevant slant height, and a rectangular pyramid has two different slant heights.
What is the volume of a pyramid with the top cut off?
A frustum needs its own formula using both the top and bottom areas and the height. It is not simply a scaled version of the full pyramid.
References
- [1]Wikipedia, Pyramid (geometry) — https://en.wikipedia.org/wiki/Pyramid_(geometry)
- [2]Wikipedia, Volume — https://en.wikipedia.org/wiki/Volume
- [3]Wikipedia, Square pyramid — https://en.wikipedia.org/wiki/Square_pyramid
- [4]Wikipedia, Frustum — https://en.wikipedia.org/wiki/Frustum