Volume and Surface Area Calculator

What this calculator answers

Choose a solid (cube, cuboid, cylinder, sphere, cone, square pyramid or rectangular pyramid) and enter its measurements in one unit. You get the volume, the total surface area, the lateral (side) area and the base area together, plus how much the solid holds in liters, US gallons and imperial gallons. For cylinders, spheres and cones the answers are also written exactly in terms of π, and every step is shown with your numbers.

It also works backward: give it a volume in gallons, liters or cubic units and the other measurements, and it finds the missing height, or the edge of a cube or the radius of a sphere.

How to use it

• Solid: the shape. A cuboid is a rectangular box. The drawing in the results follows your measurements, so you can check that each number went where you meant.
• Length unit: one unit for every measurement. Volumes come out in its cube (inches give in³) and areas in its square (in²). Convert mixed measurements first: 18 in is 1.5 ft.
• Find: Volume and area is the usual direction. Size from a volume asks for the volume and its unit and solves the one length it can’t have.
• Radius or diameter: type either one for a cylinder, sphere or cone. Typing in one box clears the other, and the line under the empty box shows its value.
• Vertical height or slant height: the same for a cone or square pyramid. The vertical height runs straight down from the tip to the center of the base; the slant height runs down the outside. A rectangular pyramid takes the vertical height only, because its two pairs of faces have different slant heights.
• Decimal places: 2 unless your homework asks for something else. “To the nearest tenth” means 1.

The Try buttons load the cases worked through below: the height that holds 5 US gallons, a cone known by its slant height, a 9-inch ball, a rectangular pyramid and a 1-foot cube in gallons.

Volume and surface area formulas for common solids

• $r$ is the radius (half the diameter) and $h$ the vertical height, at right angles to the base.
• $s$ is the slant height: for a cone from the tip to the edge of the base, for a pyramid from the tip to the middle of a base edge. $s_1$ and $s_2$ are the slant heights of a rectangular pyramid’s two pairs of faces (below).
• Lateral area is the sides only. The total adds the base, or both bases for a cube, cuboid or cylinder.

Worked example: a 22-inch rain barrel

A rain barrel measures 22 in across and 34 in tall on the inside. How much water does it hold, and how much surface is there to paint?

1. Radius: 22 ÷ 2 = 11 in.
2. Volume: V = π × 11² × 34 = 4,114π ≈ 12,924.51 in³.
3. US gallons: 12,924.51 ÷ 231 ≈ 55.95 US gal, because a US gallon is exactly 231 in³. In liters, 12,924.51 × 0.016387064 ≈ 211.79 L, which is 46.59 imperial gallons.
4. Lateral area: 2π × 11 × 34 = 748π ≈ 2,349.91 in², the curved side. Unrolled, it is a rectangle 22π ≈ 69.12 in long (the circumference) and 34 in tall.
5. Each base: π × 11² = 121π ≈ 380.13 in².
6. Total surface area: 2 × 121π + 748π = 990π ≈ 3,110.18 in².

With the lid off, painting the outside means the side and the bottom only: 748π + 121π = 869π ≈ 2,730.04 in², or about 18.96 ft² after dividing by 144. With thin walls, the outside is close enough to these inside measurements; for a thick-walled container, measure the outside for paint.

Volume of a cylinder in gallons or liters

Find the volume in cubic inches and divide by 231, or in cubic feet and multiply by 7.48. A US gallon is defined as exactly 231 in³, and a cubic foot is 1,728 in³, so it holds 1,728 ÷ 231 ≈ 7.48 US gallons. For liters, multiply cubic inches by 0.016387064 or cubic feet by 28.316846592; both factors are exact. A cube 1 ft on each side holds 7.48 US gallons, or 28.32 L.

The imperial gallon used in the UK is larger, 4.54609 L, so the barrel’s 55.95 US gallons are 46.59 imperial gallons. The calculator shows both and labels them.

A tank standing upright holds a share of its capacity equal to the share of its height that is filled, because the sides are vertical. The Capacity at other fill depths table does this for cylinders and boxes: the barrel filled to 75% (25.5 in of water) holds 41.96 US gallons.

Lateral vs total surface area

Lateral area is the area of the sides only; total surface area adds the base, or the top and bottom. Which one you need depends on the job:

• A label or wrap around a can or tank: the lateral area. A cylinder’s side unrolls into a rectangle as long as the circumference and as tall as the cylinder.
• A closed box or a ball: the total. A sphere has no base, so its whole surface counts.
• An open-topped container, painted or lined: the lateral area plus one base, as with the barrel above.

The Surface area face by face table lists every face with its size, its area and whether it counts as a base or a side, so you can add up exactly the faces your project covers.

How to find the slant height of a cone or pyramid

Use the Pythagorean theorem on the right triangle inside the solid: its sides are the vertical height, the slant height and the distance from the center of the base out to where the slant height meets the edge.

• Cone: $s = \sqrt{r^2 + h^2}$. A cone with a 5 in radius and a 13 in slant height is $\sqrt{13^2 - 5^2} = 12$ in tall. Its volume is 100π ≈ 314.16 in³, its curved side 65π ≈ 204.2 in² and its total surface area 90π ≈ 282.74 in².
• Square pyramid: $s = \sqrt{h^2 + (a/2)^2}$. With a 6 m base and a height of 4 m, $s = \sqrt{4^2 + 3^2} = 5$ m. The four faces cover 2 × 6 × 5 = 60 m², the total is 36 + 60 = 96 m², and the volume is 48 m³.
• Rectangular pyramid: two slant heights. The faces over the length edges lean in across half the width, and the faces over the width edges across half the length:

For a base 8 ft by 6 ft and a height of 4 ft, $s_1 = \sqrt{4^2 + 3^2} = 5$ ft and $s_2 = \sqrt{4^2 + 4^2} \approx 5.66$ ft. The lateral area is 2 × ½ × 8 × 5 + 2 × ½ × 6 × 5.66 ≈ 73.94 ft², and the total with the 48 ft² base is 121.94 ft². Using 5 ft for all four faces would give 70 ft², 5.3% short.

The slant height is not the edge from the tip to a corner. That lateral edge is longer, 6.4 ft for this pyramid, because it also crosses the other half of the base.

Why a cone holds one third of a cylinder

A cone holds exactly one third of a cylinder with the same base and height, and a pyramid one third of a box with the same base and height. Slice either one into thin layers parallel to the base: each layer’s width is proportional to its distance from the tip, so its area grows with the square of that distance. Adding up layers whose area grows with the square of the distance gives one third of base × height, where a cylinder’s equal layers give the whole of it.

The 5-inch cone above holds 100π in³; a cylinder with the same 5 in radius and 12 in height holds π × 5² × 12 = 300π in³.

Finding a height or radius from a volume

Convert the volume to cubic units of your measurements, then rearrange the volume formula for the missing length: $h = V \div \pi r^2$ for a cylinder, $h = 3V \div \pi r^2$ for a cone, $h = V \div lw$ for a box, $a = \sqrt[3]{V}$ for a cube and $r = \sqrt[3]{3V \div 4\pi}$ for a sphere.

How tall must a container 12 in across be to hold 5 US gallons? 5 × 231 = 1,155 in³, and 1,155 ÷ (π × 6²) ≈ 10.21 in. The same way, the 22-inch barrel reaches 55 US gallons at a depth of about 33.42 in, just under its 34 in height.

Reading the result

• The headline is the volume, with its exact π form and what it holds in liters and US gallons. In Size from a volume it is the missing length instead.
• The tiles give the total, lateral and base areas, the capacity in liters and in US and imperial gallons, and one more length: the space diagonal of a box (the longest rod that fits inside), the circumference of a cylinder or sphere, or whichever of the slant and vertical heights of a cone or pyramid you didn’t enter.
• The steps show each formula with your numbers, including the unit factor used for every conversion.
• The tables break the surface area down face by face, show an upright box or cylinder at other fill depths, and give the volume and area in the other units. Each has a CSV download.
• Save for comparison keeps up to three solids side by side, with the change in capacity between them.

How volume and area grow with size

Scale every length by the same factor and the area grows with its square, the volume with its cube. A barrel 10% wider and 10% taller has 1.1² = 1.21 times the surface, 21% more paint, and 1.1³ = 1.331 times the volume, 33.1% more water. The same rule makes a small slip in a length a large error in the volume.

Assumptions and limitations

• Right solids only. The top of a cylinder is directly above its base, and the tip of a cone or pyramid is above the center of its base. Leaning (oblique) solids have different areas.
• Capacity is the whole volume, to the brim. Leave room for a lid, a fill line or an overflow in real containers.
• Inside or outside is up to you. Capacity comes from inside measurements and paint from outside ones; the calculator doesn’t subtract a wall thickness.
• Seven solids. Hemispheres, capsules, frustums (a cone or pyramid with the top cut off) and tanks lying on their side are not covered.
• Rounding. Results are rounded for display only; answers written with π are exact for the numbers you typed.

Common mistakes

• Using the diameter as the radius. The barrel’s 22 in taken as the radius gives 223.8 US gallons, four times the true 55.95. For a sphere the error is eight times.
• Mixing inches and feet. A tank 4 ft by 2 ft by 18 in, typed as 4 × 2 × 18 in feet, gives 144 ft³ instead of 4 × 2 × 1.5 = 12 ft³ (89.77 US gallons).
• Using the slant height as the vertical height. For the cone with a 13 in slant side, 1/3 × π × 5² × 13 gives 340.34 in³, 8.3% more than the true 314.16 in³.
• Reading feet and inches as decimals. 5 ft 11 in is 5.9167 ft, not 5.11 ft. The calculator points this out when a length in feet ends in .10 or .11.
• Mixing up US and imperial gallons. A tank of 55.95 US gallons holds only 46.59 imperial gallons.
• Buying paint for the total area of an open container, or for only the lateral area of a closed one. Check which faces the job covers in the face-by-face table.