Volume of a pyramid, formula and calculator.

The volume of a pyramid is V = 1/3 × B × h, where B is the area of the base and h is the height. Take a square base of 6 cm and a height of 9 cm. The base area is 6 × 6 = 36 cm². Then V = 1/3 × 36 × 9 = 108 cm³. A box with the same base and height would hold 36 × 9 = 324 cm³, so the pyramid is exactly a third of it.

How to use the converter

1. Choose the Base: square, rectangle, or a base area you already know.
2. Type the side or sides (or the area) and the Height, and choose a Unit of length.
3. Read the big result and The working, which shows the base area first and then the third.
4. For a square or rectangle base the cards also give the slant height and the surface area. Use Copy result or Copy link to keep the answer.

Height and slant height are different

The height is the straight line from the tip down to the middle of the base, at a right angle to it. The slant height runs from the tip down the middle of a sloping face. The slant height is longer. For the 6 cm square base with a height of 9 cm, the slant height is √(9² + 3²) = √90 = 9.4868 cm, because the base of that triangle is half the side, 3 cm. Using 9.4868 in the volume formula would give 113.84 cm³ instead of 108, a gap of 5.84 cm³. A drawing may show only the slant height; if so, turn it into the height first with h = √(slant² − (half the side)²).

Why one third

Take a cube of side a. Three equal pyramids, each with one square face of the cube as its base and its tip at the same corner of the cube, fill it exactly. The cube has a volume of a³, so one pyramid has a³ ÷ 3. That is base area a² times height a, divided by 3. The same rule holds for any base and any pyramid with a flat base, and for a cone, whose base is a circle.

Square pyramids with height equal to the side

Worked examples

• A rectangle base, 6 by 4, height 9. The base area is 6 × 4 = 24. V = 1/3 × 24 × 9 = 72 cubic units.
• A base area of 36 m² and a height of 5 m. V = 1/3 × 36 × 5 = 60 m³, which is 60,000 liters.
• A square base of 100 m and a height of 60 m. The base area is 10,000 m². V = 1/3 × 10,000 × 60 = 200,000 m³. These are round example numbers.
• Surface area, square base 6 and height 9. The four sloping faces are triangles, each with area 1/2 × 6 × 9.4868 = 28.460. Four of them are 113.842. Add the base 36 to get 149.842 cm².

A pyramid with the top cut off

A frustum is a pyramid with the tip sliced off by a flat cut parallel to the base. Its volume is h ÷ 3 × (A1 + A2 + √(A1 × A2)), where A1 and A2 are the two flat areas and h is the straight height between them. Take a square base of side 6 cm, a square top of side 3 cm and a height of 9 cm. A1 = 36, A2 = 9 and √(36 × 9) = 18. So V = 9 ÷ 3 × (36 + 9 + 18) = 3 × 63 = 189 cm³. Check it another way. The top is half the width of the base, so the cut sits halfway up the full pyramid, which is 18 cm tall. The full pyramid is 36 × 18 ÷ 3 = 216 cm³, the piece removed is 9 × 9 ÷ 3 = 27 cm³, and 216 − 27 = 189 cm³.

When you know the slant height

The slant height is the length up the middle of a sloping face, and it is longer than the height. For a square base of side a, the height is √(slant² − (a ÷ 2)²). With a = 6 cm and a slant height of 15 cm, h = √(225 − 9) = √216 = 14.6969 cm. The volume is 36 × 14.6969 ÷ 3 = 176.3633 cm³. Do not put the slant height straight into the volume formula. It would give 36 × 15 ÷ 3 = 180 cm³, which is too much.

The same one third for a cone

A cone is a pyramid with a round base, and it follows the same rule: one third of the base area times the height. With a radius of 3 cm and a height of 9 cm, the base is π × 9 = 28.2743 cm² and V = 28.2743 × 9 ÷ 3 = 84.8230 cm³. Type the base area into the converter with Base area I know and you get this too.

What happens when you change one size

The height enters the volume once, so doubling the height doubles the volume. The base side enters twice, so doubling the side makes the volume four times as big. The table shows a square base with the side changing and two heights.

Mistakes to avoid

• Using the slant height as the height. It always makes the volume too big.
• Forgetting the third. Base times height is the volume of a box, three times too large.
• Using the base side as the base area. For a square, square the side first.
• Mixing units. A base in cm and a height in m must be put in one unit first.

The base area of a rectangle is the area of a rectangle. For a round solid, see the volume of a cylinder.