Volume & Geometry Calculator
The volume calculator finds the volume and surface area of common 3D shapes — cylinder, cone, sphere, cube, and rectangular prism — showing every step of the working. Enter the dimensions of your shape below and get a complete step-by-step solution instantly.
This free volume and geometry calculator also converts any result into liters, cubic meters, or weight — useful for water tanks, aquariums, and shipping containers. No signup required.
Step-by-Step Solution
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 | V = l×w×h | SA = 2(lw+lh+wh) |
| Circle (2D) | A = πr² | C = 2πr |
How to Use This Volume Calculator
- Select your shape from the dropdown: cylinder, cone, sphere, cube, rectangular prism, or circle (2D).
- Enter the required dimensions for that shape (radius, height, side length, etc.).
- Click Calculate. The calculator instantly shows the volume and surface area, with every step of the working.
- Use the Unit Converter tab to see the result in liters, cubic meters, or cubic feet — useful for tanks and containers.
- Use the Volume to Weight tab to convert a volume into mass, once you enter the density of the material or liquid.
- The SVG shape diagram updates automatically to show your shape with labeled dimensions.
The calculator accepts positive numbers and decimals for every dimension.
Worked Examples — Step by Step
Example 1: Volume and Surface Area of a Cylinder
Find the volume and surface area of a cylinder with radius 7 cm and height 10 cm. (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
Example 2: Volume and Surface Area of a Cone
Find the volume and surface area of a cone with radius 3 cm, height 4 cm, and slant height 5 cm (a 3-4-5 right triangle).
- 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
Example 3: Volume and Surface Area of a Sphere
Find the volume and surface area of a sphere with radius 6 cm.
- 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 ≈ 904.32 cm³
- Write the surface area formula: SA = 4πr²
- Substitute: SA = 4 × π × 6² = 4 × π × 36 = 144π
- Calculate: SA ≈ 452.16 cm²
Example 4: Water Tank Volume in Liters
A rectangular water tank measures 2 meters long, 1 meter wide, and 1.5 meters high. Find its capacity in liters.
- 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
Example 5: Converting Volume to Weight (Density)
Using the same 3 m³ tank from Example 4, find the weight of the water it holds. 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
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 roughly doubles or quadruples the result depending on the formula.
- 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.
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. Select your shape in the calculator above to see its exact formula.
- How do you find the volume of a cylinder?
- Use the formula 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 the formula 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 904.32 cm³.
- How do you calculate the volume of a cone?
- Use the formula V = (1/3)πr²h, where r is the radius of the circular base and h is the height. This is exactly one-third the volume of a cylinder with the same radius and height. For example, a cone with radius 3 cm and height 4 cm has a volume of 12π, about 37.70 cm³.
- 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. For example, a rectangular tank measuring 2m × 1m × 1.5m 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, using cubic units. Surface area measures the total area of all the outer faces of a shape, using square units. Volume tells you how much a container can hold; surface area tells you how much material it takes to cover it.
- How do you convert volume to weight?
- Multiply the volume by the density of the material or liquid: mass = volume × density. For example, water has a density of 1,000 kg per cubic meter, so a 3 cubic meter tank of water weighs 3,000 kg.
- What is CBM in shipping and logistics?
- CBM stands for cubic meters, and it measures how much space a package or shipment takes up. It is calculated the same way as the volume of a rectangular prism: CBM = length × width × height, with all three measurements in meters.
Related Calculators
Pythagorean Theorem CalculatorFind the slant height of a cone or pyramid — a right-triangle calculation needed for surface area formulas. Square Root Calculator
Solve for a missing radius or side length when you already know the volume and need to work backward. Percentage Calculator
Find what percentage full a tank is, or calculate a percent increase or decrease in volume.
Density Calculator
Formula: Density = Mass ÷ Volume. Solve for any one variable.