Volume & Geometry Calculator
QUICK ANSWER
Pick a shape, enter its size, and this calculator returns the volume and surface area with every step. Cube: V = a³. Sphere: V = (4/3)πr³. Cylinder: V = πr²h. Cone: V = (1/3)πr²h. Box or tank: V = l × w × h. Cone frustum: V = (1/3)πh(r² + rR + R²). Capsule: V = πr²h + (4/3)πr³. A circle has an area, πr², but no volume. Results are also given in liters, cubic meters and gallons.
This volume and geometry calculator finds the volume and surface area of eight shapes: cube, sphere, cylinder, cone, box or tank, circle, cone frustum and capsule. It shows the exact answer in terms of π, converts the result to liters, cubic meters and gallons, previews the shape in 3D, and includes a density tab for mass and volume. Every answer comes with a step-by-step solution and a check, and no signup is needed.
Calculation
Cube volume and surface area
Formula
3D previewRotates on its own and rescales to your numbers when you press Calculate
Step-by-Step Solution
Given
Formula
Step-by-step working
Check your answer
Final summary
What Is Volume?
Volume is the amount of three-dimensional space a shape takes up, measured in cubic units like cubic centimeters (cm³), cubic meters (m³), or liters. Every solid shape has its own volume formula based on its dimensions, radius, height, base, or side length. Volume is closely related to surface area, which measures the total area of all the outer faces of a shape instead of the space inside it. Cuemath's guide to volume and surface area covers additional shapes and practice problems if you want more examples.
Volume vs Surface Area
| Term | Meaning |
|---|---|
| Volume | How much space is inside a shape, measured in cubic units (cm³, m³, liters). Answers: "how much can this shape hold?" |
| Surface area | The total area of every outer face of a shape, measured in square units (cm², m²). Answers: "how much material covers this shape?" |
Volume and Surface Area Formulas
The table below covers the shapes supported by this calculator. Where: r = radius, h = height, s = side length, l = slant height, w = width. π (pi) ≈ 3.14159 (or 22/7 as a common approximation).
Formula Reference
| Shape | Volume Formula | Surface Area Formula |
|---|---|---|
| Cube | V = a³ | SA = 6a² |
| Sphere | V = (4/3)πr³ | SA = 4πr² |
| Cylinder | V = πr²h | SA = 2πr² + 2πrh |
| Cone | V = (1/3)πr²h | SA = πr(r + √(r²+h²)) |
| Rectangular Prism / Tank | V = l×w×h | SA = 2(lw+lh+wh) |
| Circle (2D) | A = πr² | C = 2πr |
| Cone Frustum | V = ⅓πh(r²+rR+R²) | SA = π(r+R)l + πr² + πR² |
| Capsule | V = πr²h + ⁴⁄₃πr³ | SA = 2πrh + 4πr² |
How to Use This Volume and Geometry Calculator
- Pick a tab: Cube, Sphere, Cylinder, Cone, Box / Tank, Circle, Frustum, Capsule, or Density.
- Choose the unit (mm, cm, m, inches or ft) and the value of π. Use exact π for the most accurate answer, or 3.14 or 22/7 to match a textbook.
- Type the dimensions. Whole numbers, decimals and fractions such as 3/4 all work, and every size must be greater than 0.
- Click Calculate or press Enter. The Step-by-Step Solution box opens directly under the inputs, and the 3D preview rescales to your shape.
- Read the volume and surface area, the exact answer in terms of π, and the capacity in liters, cubic meters and US gallons. Then follow the numbered steps and the Check your answer line.
- Use Share result or Download PDF to keep the full working as a PDF.
To reproduce the worked examples below, choose π = 22/7 for Example 1 and exact π for Examples 2 and 3.
What the Step-by-Step Solution Shows
After you press Calculate, the answer box shows the method as well as the number.
- The answer: the volume first, then the surface area, and for exact π the answer in terms of π (for example 490π) next to its decimal value.
- Key values: liters, cubic meters, US gallons, the weight if the shape were full of water, and shape extras such as slant height or the surface area of a tank with no lid.
- Numbered steps: the formula, your numbers substituted in, the simplified result, and the conversion to liters, each with a one-line reason.
- Check your answer: an independent route to the same number, such as a cone being one third of a cylinder, a frustum being a big cone minus a small cone, or Archimedes’ two thirds rule for a sphere.
- Share result and Download PDF: Share sends the answer and a link through your apps, and Download PDF gives a file for homework or a quote.
Units, Pi and Rounding
The unit you choose is used for every length, so volume comes out in cubic units of the same kind (cm³) and surface area in square units (cm²). Conversions to liters use exact length conversions (1 in = 2.54 cm and 1 ft = 30.48 cm), so there is no rounding in the conversion itself. With exact π, answers are shown both in terms of π and as a decimal rounded to 4 places. With π = 3.14 or 22/7, the arithmetic is exact and the result is shown as a decimal. Sizes can have up to 15 digits. A shape with a slant height, such as a cone or frustum, shows the slant height as an exact number when it is whole or a simple fraction, and as a rounded decimal when it is not.
Worked Examples, Step by Step
Example 1: Volume and Surface Area of a Cylinder
Marco is designing a cylindrical rain barrel with a radius of 7 cm and height of 10 cm for a school science-fair model. He needs both the volume and the amount of sheet metal (surface area) required to build it. (Using π ≈ 22/7 for a clean result.)
- Write the known values: r = 7 cm, h = 10 cm
- Write the volume formula: V = πr²h
- Substitute: V = (22/7) × 7² × 10 = (22/7) × 49 × 10
- Calculate: V = 22 × 7 × 10 = 1,540 cm³
- Write the surface area formula: SA = 2πr² + 2πrh = 2πr(r + h)
- Substitute: SA = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17
- Calculate: SA = 748 cm²
Marco's barrel holds 1.54 liters per 10 cm of height, and he'll need 748 cm² of sheet metal per section, enough to plan exactly how many sheets to buy.
Example 2: Volume and Surface Area of a Cone
Priya is 3D-printing a small decorative cone with radius 3 cm, height 4 cm, and slant height 5 cm (a 3-4-5 right triangle) and wants to know how much filament and paint she'll need.
- Write the known values: r = 3 cm, h = 4 cm, l (slant height) = 5 cm
- Write the volume formula: V = (1/3)πr²h
- Substitute: V = (1/3) × π × 3² × 4 = (1/3) × π × 9 × 4
- Calculate: V = 12π ≈ 37.70 cm³
- Write the surface area formula: SA = πr² + πrl = πr(r + l)
- Substitute: SA = π × 3 × (3 + 5) = 24π
- Calculate: SA ≈ 75.40 cm²
At roughly 1.24 g/cm³ for standard PLA filament, Priya's cone will use about 47 grams of plastic, and she now knows exactly how much surface she'll need to prime before painting.
Example 3: Volume and Surface Area of a Sphere
Owen is ordering a custom leather cover for a globe with radius 6 cm, and needs to know both the air it holds and the leather panel area required to wrap it.
- Write the known value: r = 6 cm
- Write the volume formula: V = (4/3)πr³
- Substitute: V = (4/3) × π × 6³ = (4/3) × π × 216
- Calculate: V = 288π ≈ 904.78 cm³
- Write the surface area formula: SA = 4πr²
- Substitute: SA = 4 × π × 6² = 4 × π × 36 = 144π
- Calculate: SA = 144π ≈ 452.39 cm²
Owen's globe holds about 905 cm³ of air, and the leatherworker will need to cut roughly 452 cm² of material, plus a small allowance for seams.
Example 4: Water Tank Volume in Liters
Ines just installed a rectangular water tank measuring 2 meters long, 1 meter wide, and 1.5 meters high on her rooftop, and wants to know how many liters it holds before she books a water-delivery truck.
- Write the known values: length = 2 m, width = 1 m, height = 1.5 m
- Write the volume formula for a rectangular prism: V = l × w × h
- Substitute: V = 2 × 1 × 1.5
- Calculate: V = 3 m³
- Convert cubic meters to liters: 1 m³ = 1,000 liters
- Multiply: 3 m³ × 1,000 = 3,000 liters
Ines now knows her tank holds 3,000 liters, so she can request the right delivery size instead of guessing and running short mid-week.
Example 5: Converting Volume to Weight (Density)
Ines wants to know how much her full rooftop tank from Example 4 actually weighs, since her contractor needs to confirm the roof can safely support it. Water has a density of 1,000 kg per cubic meter.
- Write the known values: volume = 3 m³, density = 1,000 kg/m³
- Write the formula: mass = volume × density
- Substitute: mass = 3 × 1,000
- Calculate: mass = 3,000 kg
At 3 metric tons when full, Ines can pass that number straight to her contractor to confirm the rooftop's load capacity before the tank is filled.
When to Use a Volume & Geometry Calculator
Volume calculations come up anywhere you need to know how much space something takes up or how much it can hold:
- Geometry class: finding the volume and surface area of standard 3D shapes for homework or exams.
- Water tanks and plumbing: calculating how many liters a cylindrical or rectangular tank holds, based on its measurements, also useful alongside the Percentage Calculator for checking what percentage full a tank is at a given fill height.
- Aquariums and ponds: sizing a fish tank or pond and figuring out how much water (and how many fish) it can support.
- Construction and materials: estimating how much concrete, soil, or gravel is needed to fill a space of a known shape.
- Shipping and logistics: converting a package's dimensions into cubic meters (CBM) for freight and container-loading calculations.
Converting Volume to Liters, Cubic Meters, and Weight
A raw volume in cubic centimeters or cubic meters isn't always the most useful number. This calculator converts any result into the units that matter for your situation.
| Conversion | How it works |
|---|---|
| Cubic units → Liters | 1 cubic meter (m³) = 1,000 liters. 1 cubic centimeter (cm³) = 0.001 liters (1 milliliter). Common for tanks, tubs, and container capacity. |
| Cubic units → Weight | Multiply volume by the material's density: mass = volume × density. Water's density is 1,000 kg/m³, so a 3 m³ tank of water weighs 3,000 kg. |
| Cubic units → CBM (shipping) | CBM simply means cubic meters, used in freight to calculate how much space a shipment takes up in a container. CBM = length × width × height, all measured in meters. |
Common Mistakes in Volume and Surface Area Calculations
- Using diameter instead of radius: most volume formulas require the radius, not the diameter, plugging in the diameter by mistake makes the radius twice too big, which multiplies a circle area, cylinder volume or cone volume by 4 and a sphere volume by 8.
- Mixing units within one calculation: combining centimeters and meters in the same formula without converting first produces a result that's off by a large, easy-to-miss factor.
- Confusing similar shape formulas: a cone's volume is exactly one-third of a cylinder with the same radius and height, forgetting that ⅓ factor is one of the most common volume errors.
- Applying surface area formulas to open containers: a water tank open at the top needs one less face counted in its surface area than the closed-shape formula assumes. The Box / Tank tab shows the surface area with no lid for this reason.
Frequently Asked Questions
- What is the formula for volume?
- There is no single formula for volume. Each shape has its own, based on its dimensions. A cube uses V = s³, a cylinder uses V = πr²h, a sphere uses V = (4/3)πr³, and a cone uses V = (1/3)πr²h. Pick your shape in the calculator above to see its exact formula and the working.
- How do you find the volume of a cylinder?
- Use V = πr²h, where r is the radius of the circular base and h is the height. For example, a cylinder with radius 7 cm and height 10 cm has a volume of (22/7) × 7² × 10 = 1,540 cm³, using π ≈ 22/7.
- How do you find the volume of a sphere?
- Use V = (4/3)πr³, where r is the radius. For example, a sphere with radius 6 cm has a volume of (4/3) × π × 6³ = 288π, which is about 904.78 cm³.
- How do you calculate the volume of a cone?
- Use V = (1/3)πr²h, where r is the radius of the base and h is the vertical height. This is exactly one third of a cylinder with the same radius and height. A cone with radius 3 cm and height 4 cm has a volume of 12π, about 37.70 cm³.
- How do you find the volume of a cone frustum?
- Use V = (1/3)πh(r² + rR + R²), where r is the top radius, R is the bottom radius and h is the height. A frustum with r = 2, R = 5 and h = 6 has a volume of (1/3) × π × 6 × 39 = 78π, about 245.04 cubic units.
- How do you find the volume of a capsule?
- A capsule is a cylinder with a half sphere on each end, so V = πr²h + (4/3)πr³, where h is the height of the cylinder part only. Add the cylinder volume to the volume of one full sphere.
- How do you calculate the volume of a water tank in liters?
- Find the volume in cubic meters using the formula for the tank's shape, then multiply by 1,000 to convert to liters. A rectangular tank measuring 2 m × 1 m × 1.5 m has a volume of 3 m³, which equals 3,000 liters.
- What is the difference between volume and surface area?
- Volume measures how much three-dimensional space is inside a shape, in cubic units. Surface area measures the total area of all the outer faces, in square units. Volume tells you how much a container holds; surface area tells you how much material covers it.
- How do you convert volume to weight?
- Multiply the volume by the density of the material: mass = volume × density. Water has a density of 1,000 kg per cubic meter, so a 3 cubic meter tank of water weighs 3,000 kg.
- How do you calculate density?
- Divide mass by volume: density = mass ÷ volume. A 540 g block with a volume of 200 cm³ has a density of 2.7 g/cm³. The Density tab can also solve for mass or volume when you know the other two values.
- Which value of pi should I use?
- Use the exact π option for the most accurate answer. Use 3.14 or 22/7 when your teacher or textbook asks for them, because the answers will then match the textbook exactly. The calculator shows the exact answer in terms of π as well.
- What is CBM in shipping and logistics?
- CBM stands for cubic meters and measures how much space a shipment takes up. It is calculated like the volume of a box: CBM = length × width × height, with all three in meters.
Density, Mass and Volume
Density links the volume you just calculated to a weight. The formula is density = mass ÷ volume, and it can be rearranged to find mass (mass = density × volume) or volume (volume = mass ÷ density). The Density tab solves for whichever of the three you leave blank. Use g and cm³ for density in g/cm³, or kg and m³ for density in kg/m³. Note that 1 g/cm³ equals 1,000 kg/m³.
| Material | Density (g/cm³) | Density (kg/m³) |
|---|---|---|
| Water (about 4 °C) | 1.00 | 1,000 |
| Ice | 0.92 | 920 |
| Aluminium | 2.70 | 2,700 |
| Concrete (typical) | 2.40 | 2,400 |
| Iron | 7.87 | 7,870 |
| Copper | 8.96 | 8,960 |
| Lead | 11.34 | 11,340 |
| Gold | 19.32 | 19,320 |
Values are typical reference figures and vary slightly with temperature, purity and alloy. A material with a density above 1 g/cm³ sinks in water, and one below 1 g/cm³ floats.