Volume Calculator

Every solid, three-dimensional object occupies space. Whether it’s a smartphone resting on a desk, a water storage tank in your neighborhood, or a basketball on a court, they all take up room.

In mathematics and science, this occupied space is defined as volume. Volume can also refer to an object's capacity. For instance, instead of focusing on the physical space a water container takes up in a garage, we often want to know its capacity—the exact amount of water it can hold.

Calculating volume is an essential skill across numerous scientific and mathematical disciplines.

Our comprehensive volume calculator simplifies this process by supporting multiple measurement units and a wide variety of 3D shapes. Better yet, the calculator doesn't just give you the final answer; it displays the exact volume formula and walks you through a step-by-step calculation process. In this guide, we will explore how to calculate volume, explain the formulas for various geometric shapes, and walk through practical, real-world examples.

Units and Measurements

To ensure accuracy and reliability, volume calculations rely on standard units of measurement. The standard SI (International System of Units) unit for volume is the cubic meter (m³). However, the volume of smaller objects is often expressed in smaller units, such as cubic centimeters (cm³) or cubic millimeters (mm³).

Depending on your specific needs, you may prefer one measurement system over another. Our volume calculator fully supports both the Metric System and Imperial/US Customary Units. You have the complete freedom to choose from the following units:

• kilometers,
• meters,
• centimeters,
• millimeters,
• micrometers,
• nanometers,
• angstroms,
• miles,
• yards,
• feet,
• inches.

When using manual formulas to calculate volume, you must work with homogeneous units. Typically, this means converting all measurements to the exact same unit to simplify the math. For example, to calculate the volume of a cylinder with a height of 75 cm and a radius of 0.5 m, you would either convert the height to meters (yielding a result in cubic meters) or convert the radius to centimeters (yielding a result in cubic centimeters).

But what if you want to input the height in inches and the radius in nanometers? Our calculator handles this seamlessly, performing the necessary unit conversions behind the scenes and clearly displaying the steps.

You can select a different unit for every single measurement input, and the volume formula calculator will still return a highly accurate result. Let’s say you have a cylinder with a height of 5 inches and a radius of 10,506,070 nanometers. Simply navigate to the cylinder volume calculator section, enter the values, and select the corresponding units from the dropdown menus.

The calculator will immediately output the volume in two formats: 2.6874044006564 inches³ (cubic inches) and 4.4038667907438E+22 nanometers³ (cubic nanometers). It provides both options because it assumes you want your final answer in one of the base units you provided. The tool even displays the complete calculation process alongside the unit conversion!

The Volume Calculator: Scope, Features, and Examples

The methods used to calculate volume vary heavily depending on the figure. Many standard geometric shapes rely on straightforward arithmetic formulas based on properties like edge length or radius.

Other shapes are significantly more complex, making direct volume calculation impossible. In those cases, advanced computational methods—such as geometrical integration and finite element analysis—are required. Fortunately, our volume calculator supports a massive range of objects, making it incredibly easy to find the volume of almost anything.

Sphere

A sphere is the perfect three-dimensional equivalent of a circle. A classic example is any completely round ball (like a baseball or a globe). The volume formula for a sphere is:

$$V_{sphere}=\frac{4}{3}π r^3$$

As you can see, the volume of a sphere depends entirely on its radius (r). The radius is defined as the exact distance from the center of the sphere to any point on its outer surface. Given that a standard baseball has a radius r = 3.65 cm, we can use our sphere volume calculator to find its volume:

$$Volume = \frac{4}{3}πr^3 = \frac{4}{3} × π × 3.65^3 = 203.68882488692 \ centimeters^3$$

Cone

A cone is a 3D shape consisting of a circular base that tapers smoothly to a single vertex point, known as the apex. All points on the circumference of the base are connected to this apex by straight line segments. We define a cone's properties using two primary measurements: the radius of the circular base (r) and the height from the center of the base to the apex (h).

A cone's volume can be expressed as:

$$V_{cone}=\frac{1}{3}{π r}^2h$$

r is the radius, and h is the height of the cone

Imagine you're hosting a birthday party and want to make DIY cone-shaped party hats that can double as popcorn holders later in the evening.

If you design cone hats with a radius of 7.5 cm and a height of 0.45 m, you can use the cone volume calculator to determine exactly how much space is inside each hat.

0.45 meters = 45 centimeters

$$Volume = \frac{1}{3}πr^2h = \frac{1}{3} × π × 7.52^2 × 45 = 2650.7188014664 \ centimeters^3$$

This gives you precisely the amount of popcorn you can fit into each cone at the end of the party!

Cube

Who hasn't tried solving a Rubik's Cube?

A cube is a geometric object featuring 8 vertices and 6 perfectly equal square sides. A cube's volume relies on a single measurement: the length of the cube's side (a).

$$V_{cube}=a^3$$

Suppose we want to purchase 30 Rubik's Cubes for a youth developmental center to help children improve their cognitive abilities. We head to the store and find the perfect cubes. The length of one side of the cube is 5.7 centimeters. However, the store clerk only has one box available to transport all of the cubes. The box is completely cubic, with a side length of 20 centimeters. Will all 30 of our cubes fit inside?

The volume of the cubes:

$$Volume = 5.7³ = 185.19\ centimeters³$$

The total volume of 30 cubes would be:

$$185.19 × 30 = 5,555.7\ centimeters³$$

The volume of the box:

$$Volume = 20³ = 8,000\ centimeters³$$

By comparing the total volume of the 30 cubes to the overall volume of the box, we can see:

$$5,555.7 < 8,000$$

It turns out the cubes will fit perfectly into the box!

Cylinder

A cylinder is a 3D geometric prism featuring a uniform circular base. You can think of it as multiple identical circles stacked directly on top of one another. Like a cone, a cylinder's properties are defined by its circular radius (r) and its height (h), which is the distance from the bottom surface to the top surface. The formula for the volume of a cylinder is:

$$V_{cylinder}=π r^2h$$

Let's calculate the volume of a decorative cylindrical candle to find out exactly how much paraffin wax a craftsman needs to mold it. The planned height of the candle is 15 centimeters, and its diameter is 8 centimeters. From the diameter, we easily deduce that the radius is 4 centimeters. Using the formula, we calculate:

$$Volume = πr^2h = π × 4^2 × 15 = 240π = 753.98223686155\ centimeters^3$$

Rectangular Tank

A rectangular tank (or rectangular prism) is a variation of a cube where all adjoining edges are perpendicular, though not necessarily equal in length. This shape requires three measurements: length (l) and width (w)—which define its two-dimensional rectangular base—and height (h), which gives it a three-dimensional depth. The volume of a rectangular tank is calculated as:

$$V_{rectangular\ tank}=l × w × h$$

A classic, universal example of a rectangular tank is a standard shipping container. According to ISO standards, common shipping container measurements are:

• Width = 2.43 m
• Height = 2.59 m
• Length = 6.06 m or 12.2 m

Because these measurements are standardized globally, their volumes are as well. Go ahead and plug these dimensions into our rectangular tank volume calculator. Let's perform the calculations for both standard lengths: 6.06 m and 12.2 m.

$$Volume = 6.06 × 2.43 × 2.59 = 38.139822\ meters³$$

and

$$Volume = 12.2 × 2.43 × 2.59 = 76.78314\ meters³$$

More complex three-dimensional geometric shapes

Often, everyday objects are combinations of basic geometric shapes. For instance, what is the total volume of the figure below?

Looking closely, we can see that this object is a composite: it consists of a basic cylinder with a cone placed perfectly on top. Therefore, the total volume of the object is simply the sum of the cylinder's volume and the cone's volume:

$$V_{object}=V_{cylinder}+V_{cone}$$

Both the cylinder and the cone share a diameter of 4 cm. Knowing this, we conclude that:

$$r_{cylinder}=r_{cone}=\frac{4}{2}=2\ cm$$

Furthermore, the total height is a combination of both individual heights:

$$h_{object}=h_{cylinder}+h_{cone}$$

Given that:

$$h_{object}=10\ cm$$

and:

$$h_{cone}=3\ cm$$

we can easily determine that the cylinder's height is:

$$h_{cylinder}=7\ cm$$

We can now plug these values straight into the volume calculator:

$$V_{object}=V_{cylinder}+V_{cone}=87.96\ cm^3+12.56\ cm^3$$

$$V_{object}=100.52\ cm^3$$

This composite approach helps you better understand the diverse, advanced shapes our volume calculator supports below.

Capsule

A capsule is one of the most common shapes for medical pills. Using the logic from our previous example, we can see that a capsule is essentially a cylinder capped with two identical hemispheres on opposite ends.

Since two matching hemispheres make up one complete sphere, we can state that the total volume of a capsule is simply the volume of its central cylinder plus the volume of one sphere.

$$V_{capsule} = πr^2h + \frac{4}{3}πr^3 = πr^2(\frac{4}{3}r + h)$$

Where r is the radius and h is the height of the cylindrical portion.

Thanks to our dedicated capsule volume calculator, you don't need to manually calculate and combine the cylinder and sphere volumes. You can directly input the height and radius, and the tool will instantly output the accurate capsule volume.

Pharmaceutical scientists rely heavily on these calculations to design appropriately sized medication. Because a capsule must hold a highly precise dosage, scientists frequently adjust the height and radius to hit an exact target volume.

Spherical Cap

In the previous example, we noted that a hemisphere is exactly half of a sphere. A spherical cap, however, is a portion of a sphere cut off by a flat plane. A hemisphere is simply a special case of a spherical cap where that plane cuts exactly through the center.

The figure below demonstrates a typical spherical cap. In this model, (r) is the radius of the base of the cap, (R) is the radius of the full sphere, and (h) is the height of the cap. Because these variables are mathematically linked, knowing just two of them allows you to calculate the third!

• Given r and R; $h=R±\sqrt{R^2+r^2}$
• Given r and h; $R=\frac{h^2+r^2}{2h}$
• Given R and h; $r=\sqrt{2Rh\ -h^2}$

where:

• r is the radius of the base,
• R is the radius of the sphere,
• h is the height of the spherical cap.

The volume of a spherical cap is calculated as:

$$V_{spherical\ cap}=\frac{1}{3}π h^2(3R-h)$$

Our tool only requires two of these variables to work. For instance, if you input R = 1m and r = 0.25m, the calculator will surprisingly return two possible volumes: 0.00313 m³ and 4.1856 m³. Why?

Recalling the mathematical relationship:

$$h=R±\sqrt{R^2+r^2}$$

we see that given the values of R and r, the height (h) actually has two potential values:

$$h_1=R+\sqrt{R^2+r^2}$$

and

$$h_2=R-\sqrt{R^2+r^2}$$

This mathematical split explains why you get two different valid volumes depending on whether you use $h_1$ or $h_2$.

Note: The rule R ≥ r must always hold true. If you accidentally input a base radius larger than the ball radius, the calculator will helpfully return an error message to let you know the measurements were mixed up.

Conical Frustum

You can create a conical frustum by slicing the top off a cone with a perfectly horizontal cut, parallel to its base. This leaves you with a 3D object featuring two parallel, circular surfaces of different sizes.

The volume of a conical frustum is defined as:

$$V_{conical\ frustum}=\frac{1}{3}π h(r^2+rR+R^2)$$

Where h is the height between the center of the bottom and top surfaces, r is the radius of the top surface, and R is the radius of the bottom surface (where R ≥ r).

Imagine visiting a high-end bakery and ordering a chocolate lava cake that boasts a core made of exactly "35% melted chocolate."

If you're a mathematics enthusiast, you might want to test that claim! First, measure the top radius, the bottom radius, and the total height to find the overall volume of the cake.

Suppose your measurements are r = 16 cm, R = 20 cm, and h = 10 cm.

By plugging these values into our conical frustum volume calculator, you get:

$$Volume=\frac{1}{3}π h(r^2+rR+R^2)=\frac{1}{3}π 10(16^2+16×20+20^2)= 10220.648099679 \ centimeters^3$$

To find out how much gooey goodness is inside, calculate 35% of 10,220.65 cm³. You'll find there is approximately 3,577.23 cm³ of chocolate in your cake!

Ellipsoid

When a perfect sphere is stretched or deformed in one or more directions, it creates an ellipsoid. Think of an ellipsoid as a stretched out, oval-like sphere where the distances from the center to the surface vary depending on the direction.

An ellipsoid has three distinct axes, and its volume is determined by the three radii stretching from the center to the edge of each axis. These three radii are denoted by the variables a, b, and c.

We often think of balls as perfect spheres, but ellipsoidal balls are incredibly common in sports—just look at a rugby ball! Let's assume the radii of a standard rugby ball are a = 9.3 cm, b = 9.3 cm, and c = 14.3 cm.

The volume of an ellipsoid formula is:

$$V_{ellipsoid}=\frac{4}{3}π abc$$

(Note: The order of a, b, and c doesn't matter; multiplying them in any order will yield the same result.)

Using our ellipsoid volume calculator, finding the exact volume of the rugby ball is a breeze:

$$Volume=\frac{4}{3}π abc=\frac{4}{3}× π × 9.3 × 9.3 × 14.3 = 5180.7250468112 \ centimeters^3$$

Square Pyramid

Mentioning pyramids instantly brings to mind the ancient, monolithic structures of Egypt. A square pyramid features a perfect square base that tapers up to a single apex point, connecting all four corners of the base directly to the top. The volume formula is:

$$V_{squared\ pyramid}=\frac{1}{3}a^2h$$

Here, a represents the edge length of the square base, while h is the height from the center of the base straight up to the apex.

Let's look at the majestic Great Pyramid of Khufu based on its original dimensions: h = 146.6 m and a = 230.33 m. Using our square pyramid volume calculator, we can determine its monumental size:

$$Volume=\frac{1}{3}a^2h = \frac{1}{3}230.33^2 × 146.6 = 2,592,469.9482467\ meters^3$$

Tube

Unlike a solid cylinder, a tube is completely hollowed out, meaning it features both an outer diameter and an inner diameter. To find the exact volume of the material making up the tube, you have to account for the difference between these two diameters.

$$V_{tube}=π\frac{d_1^2-d_2^2}{4}l$$

As you might have guessed, d₁ and d₂ represent the outer and inner diameters of the tube, respectively, while l represents the tube's total length.

Let's use this formula to figure out the volume of a concrete ring needed for a new well on a cottage property. The height (or length) of our ring is 0.89 meters, the outer diameter is 1.16 meters, and the inner diameter is exactly 1 meter.

Plugging this into our tube volume calculator gives us:

$$Volume=π\frac{1.16^2-1^2}{4} × 0.89 = 0.076896 π = 0.24\ meters^3$$