Volume Calculator
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Sphere
Unlock the power of precision with our comprehensive volume calculator the ultimate digital tool designed to simplify complex spatial calculations in seconds. As more people find this calculator to be an essential resource, it continues to remove the guesswork for students, architects, and DIY enthusiasts measuring 3D objects. By integrating advanced formulas for a wide array of shapes including spheres, cylinders, cones, cubes, pyramids, capsules, ellipsoids, frustums, and rectangular tanks we provide a one-stop solution for any volume-related task.
How to Use the Volume Calculator for Precise 3D Measurements
To use this calculator is a seamless process designed to provide instant accuracy for students and professionals alike. To begin, simply select your 3D shape from our interactive sidebar. Whether you are calculating a standard cube, a cylinder, or more complex geometries like a frustum or ellipsoid, the tool dynamically updates to show the correct mathematical formula.
Once your shape is selected, choose your preferred measurement unit from the dropdown menu. Our calculator supports global standards, including meters (m), centimeters (cm), feet (ft), and inches (in). Enter your dimensions—such as the radius (r), height (h), or length (l)—into the designated input fields.
Pro Tip: Tool use formula for the volume of sphere V = 4⁄3 πr3 to ensure geometric precision.
The calculated volume will appear instantly in the results panel. To save time on documentation, use the "Copy Full Analysis" feature to grab a detailed breakdown of your data, or click "Share Results" to send your specific calculation directly to a teammate or classmate.
Related Calculators
Table of Contents
- What is Volume?
- How to Calculate Volume?
- Volume of a Sphere
- Volume of a Cylinder
- Volume of a Cone
- Volume of a Cube
- Volume of a Pyramid
- Volume of a Capsule
- Volume of an Ellipsoid
- Volume of a Frustum
- Volume of a Rectangular Tank
- Volume Formula Table
- Volume Unit Conversion Table
- Frequently Asked Questions
What is Volume?
In physics and geometry, volume is the measurement of the amount of three-dimensional space occupied by an object. It represents the capacity of a container or the space taken by a solid object.
Volume is commonly measured using cubic units such as:
- Cubic meters (m³)
- Cubic centimeters (cm³)
- Cubic inches (in³)
- Liters (L)
For example, if a water tank has a volume of 1000 liters, it means the tank can store 1000 liters of liquid.
How to Calculate Volume
Volume calculation is a fundamental skill in science, engineering, and architecture. Whether you are determining the capacity of a container or the displacement of an object, the method you use depends entirely on the 3D geometry of the shape.
In physics, volume is the measure of three-dimensional space occupied by a solid, liquid, or gas. While basic shapes like cubes use a simple length × width × height formula, more complex curved objects require constant values like Pi (π).
To calculate volume, you must use the correct formula based on the shape of the object. Different 3D shapes require different formulas.
For example:
-
Cube / Rectangular Prism:
Volume = l × w × h(Length × Width × Height) -
Cylinder:
Volume = πr²h(Pi × Radius squared × Height) -
Sphere:
Volume = (4/3)πr³(Four-thirds × Pi × Radius cubed) -
Cone:
Volume = (1/3)πr²h(One-third × Pi × Radius squared × Height)
Using a Volume Calculator for Accuracy
For complex dimensions or professional projects, using an online volume calculator is the most efficient method. These tools reduce human error by automatically applying the correct geometric constants once you input your dimensions (radius, height, or side length).
Volume of a Sphere
Before volume of sphere we need to understand what is a sphere? A sphere is a perfectly symmetrical 3D shape where every point on its surface is exactly the same distance (the radius) from its center point. Unlike a circle, which is flat, a sphere occupies volume in three dimensions.
Examples include balls, bubbles, and planets.
The Sphere Volume represents the total 3D space contained within the sphere's surface. Think of it as the amount of water needed to fill a perfectly round ball.
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- r = Radius (distance from center to edge)
Visual Representation of 3D Sphere
Volume = 0.00
2. Step-by-Step Calculation
Calculate the volume for a sphere with a radius of 5 cm:
- Identify Radius: r = 5
- Cube the Radius: 5 × 5 × 5 = 125
- Multiply by Pi (π): 125 × 3.14159 = 392.69
- Multiply by 4/3: 392.69 × 1.333 = 523.59
- Final Result: 523.6 cm3
3. Important Tips
- The Diameter Trap: Textbooks often give the diameter. Always divide it by 2 to get the radius before using the formula.
- Cubic Units: Always label your answer with "cubed" units (e.g., m3, in3).
- Order of Operations: Always calculate r3 (the exponent) before doing any multiplication.
4. Applied Example: The Industrial Fuel Reservoir
An offshore energy company is designing a subsea fuel reservoir. To withstand high deep-sea pressure, the engineers chose a perfectly spherical design with an internal radius of 4.5 meters. Calculate the total storage capacity of this reservoir.
1. Identify the Knowns:
Radius (r) = 4.5 m
Pi (π) ≈ 3.14159
2. Apply the Standard Formula:
V = 4⁄3 π r3
3. Step by Step Calculation:
- • Cube the Radius: 4.5 × 4.5 × 4.5 = 91.125
- • Multiply by π: 91.125 × 3.14159 = 286.277
- • Multiply by 4⁄3: 286.277 × 1.333 = 381.70
Final Storage Capacity:
Volume ≈ 381.70 m3
Volume of a Cylinder
To understand the cylinder's volume, we first define the shape. A cylinder is a three-dimensional solid with two parallel, congruent circular bases connected by a curved surface. It is one of the most common shapes in the physical world.
Examples include soda cans, pipes, batteries, and candles.
The Cylinder Volume represents the total 3D space contained within the object. To calculate volume for this shape, you must first determine the area of a circle at the base (A = πr²) and then multiply that result by the perpendicular height (h) of the cylinder.
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- r = Radius of the circular base
- h = Height of the cylinder
Visual Representation of 3D Cylinder
Volume = 50.27
2. Step-by-Step Calculation
Calculate the volume for a cylinder with a radius of 3 cm and a height of 10 cm:
- Identify Dimensions: r = 3, h = 10
- Square the Radius: 3 × 3 = 9
- Multiply by Pi (π): 9 × 3.14159 = 28.27
- Multiply by Height: 28.27 × 10 = 282.7
- Final Result: 282.7 cm3
3. Important Tips
- Base Area First: Think of the formula as Base Area (πr²) × Height. This makes it easier to remember.
- Uniform Units: Ensure both radius and height are in the same unit (e.g., both cm or both inches) before calculating.
- Internal Capacity: For liquids, 1,000 cubic centimeters (cm3) is exactly equal to 1 Liter.
4. Applied Example: The Chemical Storage Drum
A manufacturing plant uses standard industrial drums to store liquid raw materials. Each drum has an internal radius of 0.3 meters and a height of 0.9 meters. Calculate how much liquid one drum can hold.
1. Identify the Knowns:
Radius (r) = 0.3 m
Height (h) = 0.9 m
Pi (π) ≈ 3.14159
2. Apply the Standard Formula:
V = π × r2 × h
3. Step-by-Step Calculation:
- • Square the Radius: 0.3 × 0.3 = 0.09
- • Multiply by π: 0.09 × 3.14159 = 0.2827
- • Multiply by Height: 0.2827 × 0.9 = 0.2544
Total Drum Capacity:
Volume ≈ 0.254 m3
Volume of a Cone
A cone is a distinctive three-dimensional shape that tapers smoothly from a flat, circular base to a single point called the apex or vertex. It is essentially a pyramid with a circular cross-section.
Examples include ice cream cones, traffic cones, party hats, and some types of volcanoes.
The Cone Volume represents the capacity of the interior space. An interesting mathematical fact is that a cone's volume is exactly one-third of the cylinder volume with the same base radius and height.
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- r = Radius of the circular base
- h = Vertical dimension height (from the center of the base to the apex)
Visual Representation of 3D Cone
Volume = 16.76
2. Step-by-Step Calculation
Calculate the volume for a circular cone with a radius of 4 cm and a height of 9 cm:
- Identify Dimensions: r = 4, h = 9
- Square the Radius: 4 × 4 = 16
- Multiply by Height: 16 × 9 = 144
- Multiply by Pi (π): 144 × 3.14159 = 452.39
- Divide by 3: 452.39 ÷ 3 = 150.80
- Final Result: 150.8 cm3
3. Important Tips
- Don't Forget the 1/3: The most common mistake is forgetting to divide the final result by 3.
- Vertical vs. Slant Height: Always use the vertical height (h), not the "slant height" (the side length), in the volume formula.
- Radius vs. Diameter: If the problem gives you the width across the base, divide it by 2 to get the radius.
4. Applied Example: The Sand Pile
A construction crew pours a load of sand that forms a perfect cone on the ground. The pile has a radius of the base is 2 meters and a height of the cone is 1.5 meters. How many cubic meters of sand are in the pile?
1. Identify the Knowns:
Radius (r) = 2 m
Height (h) = 1.5 m
Pi (π) ≈ 3.14159
2. Apply the Standard Formula:
V = 1⁄3 × π × r2 × h
3. Step-by-Step Calculation:
- • Square the Radius: 2 × 2 = 4
- • Multiply by Height: 4 × 1.5 = 6
- • Multiply by π: 6 × 3.14159 = 18.849
- • Divide by 3: 18.849 ÷ 3 = 6.283
Total Sand Volume:
Volume ≈ 6.28 m3
Volume of a Cube
A cube is a unique three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Because every edge of a cube is the same length, it is the simplest 3D shape to measure.
Examples include dice, sugar cubes, ice cubes, and Rubik's cubes.
The Cube Volume measures the total space occupied by the object. Since the length, width, and height are all equal, the calculation involves simply "cubing" the length of one side.
1. Formula
- V = Volume
- s = Side length (also known as the edge)
Visual Representation of 3D Cube
Volume = 27.00
2. Step-by-Step Calculation
Calculate the volume for a cube with a side length of 4 cm:
- Identify the Side Length: s = 4
- Set up the Equation: V = 4 × 4 × 4
- Calculate: 4 × 4 = 16, then 16 × 4 = 64
- Final Result: 64 cm3
3. Important Tips
- All Sides are Equal: You only need one measurement to find the volume of a cube. If the length, width, and height are different, it is a rectangular prism, not a cube.
- Units Matter: Because you are multiplying three dimensions (length × width × height), the units must always be cubed (e.g., m3).
- Surface Area vs. Volume: Don't confuse volume (s3) with surface area (6s2). Volume is the space inside; surface area is the space on the outside.
4. Applied Example: The Shipping Box
A logistics company uses specialized cubic containers to transport fragile electronics. Each container has an internal **side length of 0.8 meters**. Calculate the total volume available for packing inside one container.
1. Identify the Knowns:
Side (s) = 0.8 m
2. Apply the Standard Formula:
V = s3
3. Step-by-Step Calculation:
- • Multiply first two sides: 0.8 × 0.8 = 0.64
- • Multiply by the third side: 0.64 × 0.8 = 0.512
Total Shipping Volume:
Volume ≈ 0.512 m3
Volume of a Pyramid
A pyramid is a three-dimensional polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. While there are many types of pyramids, the most common is the square pyramid.
Examples include the Great Pyramids of Giza, metronomes, and certain tent designs.
The Pyramid Volume describes the amount of space inside the structure. Similar to a cone, the volume of a pyramid is exactly one-third of the volume of a prism that has the same base and height.
1. Formula
- V = Volume
- l = Length of the base
- w = Width of the base
- h = Vertical height (from the center of the base to the apex)
Visual Representation of 3D Pyramid
Volume = 12.00
2. Step-by-Step Calculation
Calculate the volume for a square pyramid with a base length of 6 cm and a height of 10 cm:
- Identify Dimensions: l = 6, w = 6, h = 10
- Calculate Base Area: 6 × 6 = 36
- Multiply by Height: 36 × 10 = 360
- Divide by 3: 360 ÷ 3 = 120
- Final Result: 120 cm3
3. Important Tips
- The "One-Third" Rule: Just like a cone is 1/3 of a cylinder, a pyramid is 1/3 of a box (rectangular prism) with the same base dimensions.
- Vertical Height vs. Slant Height: Use the straight vertical height (from the tip to the floor). Do not use the length of the triangular sides (slant height) for volume.
- Any Polygon Base: While we used a square base (l × w), the formula works for any base: V = 1⁄3 × Base Area × h.
4. Applied Example: The Glass Skylight
An architect is designing a pyramid-shaped glass skylight for a museum. The square base of the skylight has a **length of 3 meters**, and the peak is **2.5 meters high**. How much air space is contained within this skylight?
1. Identify the Knowns:
Length (l) = 3 m
Width (w) = 3 m (Square Base)
Height (h) = 2.5 m
2. Apply the Standard Formula:
V = 1⁄3 × (l × w) × h
3. Step-by-Step Calculation:
- • Calculate Base Area: 3 × 3 = 9
- • Multiply by Height: 9 × 2.5 = 22.5
- • Divide by 3: 22.5 ÷ 3 = 7.5
Total Skylight Volume:
Volume = 7.5 m3
Volume of a Capsule
A capsule is a three-dimensional geometric shape consisting of a cylinder with hemispherical ends (half-spheres). It is also known as a spherocylinder. Because it combines two different shapes, its volume is the sum of the volume of the central cylinder and the two ends.
Examples include medical pills, propane tanks, and some types of underwater research vessels.
The Capsule Volume is calculated by adding the volume of a sphere (formed by the two ends) to the volume of the middle cylinder. This makes it a composite shape calculation.
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- r = Radius of the capsule (both the cylinder and the spheres)
- h = Height of the cylindrical part only (not the total length)
Visual Representation of 3D Capsule
Volume = 35.34
2. Step-by-Step Calculation
Calculate the volume for a capsule with a radius of 2 cm and a cylinder height of 5 cm:
- Identify Dimensions: r = 2, h = 5
- Calculate Sphere Part: (4/3) × π × 23 = 33.51 cm3
- Calculate Cylinder Part: π × 22 × 5 = 62.83 cm3
- Add Together: 33.51 + 62.83 = 96.34
- Final Result: 96.34 cm3
3. Important Tips
- Total Length vs. Cylinder Height: Be careful! If a problem gives you the total length of the capsule, you must subtract two radii (one for each end) to find the cylinder height (h).
- One Sphere: Remember that two hemispheres equal one full sphere. The formula is just Sphere Volume + Cylinder Volume.
- Consistent Radius: The radius for the spherical ends and the cylindrical body must be the same for it to be a standard capsule.
4. Applied Example: The Pharmaceutical Pill
A pharmaceutical company is designing a new vitamin capsule. The medicine is contained in a capsule with a **radius of 3 mm** and a central **cylindrical length of 10 mm**. What is the total volume of medicine the capsule can hold?
1. Identify the Knowns:
Radius (r) = 3 mm
Cylinder Height (h) = 10 mm
Pi (π) ≈ 3.14
2. Apply the Combined Formula:
V = (π × r2 × h) + (4⁄3 × π × r3)
3. Step-by-Step Calculation:
- • Cylinder Volume: 3.14 × 32 × 10 = 282.6 mm3
- • Sphere Volume: 1.33 × 3.14 × 33 = 113.04 mm3
- • Total Volume: 282.6 + 113.04 = 395.64
Total Pill Capacity:
Volume ≈ 395.64 mm3
Volume of an Ellipsoid
An ellipsoid is a three-dimensional geometric figure that is the surface equivalent of an ellipse in two dimensions. While a sphere is perfectly round, an ellipsoid is "stretched" or "squashed" along its axes. If all three axes of an ellipsoid are equal, it becomes a perfect sphere.
Examples include American footballs, watermelons, and the Earth itself (which is technically an oblate spheroid).
The Ellipsoid Volume formula is very similar to the sphere formula, but instead of using a single radius cubed ($r^3$), it uses the product of the three different radii (semi-axes).
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- a, b, c = The semi-axes (lengths from the center to the surface along the x, y, and z axes)
Visual Representation of 3D Ellipsoid
Volume = 49.76
2. Step-by-Step Calculation
Calculate the volume for an ellipsoid where a = 3 cm, b = 2 cm, and c = 4 cm:
- Identify the Semi-Axes: a = 3, b = 2, c = 4
- Multiply the Axes: 3 × 2 × 4 = 24
- Multiply by Pi (π): 24 × 3.14159 = 75.40
- Multiply by 4/3: 75.40 × 1.333 = 100.53
- Final Result: 100.53 cm3
3. Important Tips
- Semi-Axis vs. Full Axis: The formula uses the semi-axes (distance from center to edge). If you have the total length across the object, divide it by 2 before calculating.
- The Sphere Connection: If a, b, and c are all the same number, you are just using the sphere formula (r × r × r).
- Unit Consistency: Make sure all three measurements (a, b, c) are in the same units (e.g., all meters) before you begin.
4. Applied Example: The Watermelon Volume
A gardener grows an oval-shaped watermelon. To estimate its weight, they first find its volume. The watermelon has semi-axes of **15 cm, 10 cm, and 10 cm**. What is its total volume?
1. Identify the Knowns:
a = 15 cm, b = 10 cm, c = 10 cm
Pi (π) ≈ 3.14
2. Apply the Standard Formula:
V = 4⁄3 × π × (a × b × c)
3. Step-by-Step Calculation:
- • Multiply Semi-Axes: 15 × 10 × 10 = 1,500
- • Multiply by π: 1,500 × 3.14 = 4,710
- • Multiply by 4/3: 4,710 × 1.33 = 6,264.3
Total Watermelon Volume:
Volume ≈ 6,264.3 cm3
Volume of a Frustum
A frustum is the portion of a solid (usually a cone or pyramid) that remains after its upper part has been cut off by a plane parallel to its base. In common terms, it is a cone with the top chopped off, creating two circular bases of different sizes.
Examples include paper coffee cups, lamp shades, flower pots, and buckets.
The Frustum Volume formula is more complex because it must account for the ratio between the top circle and the bottom circle. It is essentially the volume of the original large cone minus the volume of the smaller cone that was removed.
1. Formula
- V = Volume
- π (Pi) ≈ 3.14159
- h = Vertical height (distance between the two bases)
- r1 = Radius of the top base
- r2 = Radius of the bottom base
Visual Representation of 3D Frustum
Volume = 29.32
2. Step-by-Step Calculation
Calculate the volume for a frustum where r1 = 2 cm, r2 = 4 cm, and h = 6 cm:
- Square the Radii: r12 = 4, r22 = 16
- Multiply the Radii: 2 × 4 = 8
- Sum the Parts: 4 + 16 + 8 = 28
- Multiply by Pi and Height: 28 × 3.14159 × 6 = 527.78
- Divide by 3: 527.78 ÷ 3 = 175.93
- Final Result: 175.93 cm3
3. Important Tips
- Vertical Height Only: Ensure you are using the straight vertical distance between the two circles, not the "slant height" of the side.
- Check the Order: In the formula, it doesn't matter which radius is r1 or r2, as long as you use both.
- The Cone Test: If one of the radii is zero (r1 = 0), the formula simplifies back to a standard cone formula.
4. Applied Example: The Coffee Cup
A cafe uses paper cups that are shaped like a frustum. The top radius is **4 cm**, the bottom radius is **3 cm**, and the height is **10 cm**. How much liquid can the cup hold?
1. Identify the Knowns:
r1 = 4 cm, r2 = 3 cm, h = 10 cm
Pi (π) ≈ 3.14
2. Apply the Standard Formula:
V = 1⁄3 × π × h × (r12 + r22 + (r1r2))
3. Step-by-Step Calculation:
- • Calculations: 42(16) + 32(9) + (4×3)(12) = 37
- • Multiply by π and Height: 37 × 3.14 × 10 = 1,161.8
- • Divide by 3: 1,161.8 ÷ 3 = 387.27
Total Cup Capacity:
Volume ≈ 387.27 cm3
Volume of a Rectangular Tank
A rectangular tank (or rectangular prism) is a solid three-dimensional object which has six faces that are all rectangles. It is defined by three different dimensions: length, width, and height. It is the most common shape for containers, buildings, and swimming pools.
Examples include fish tanks, shipping containers, bricks, and cereal boxes.
The Rectangular Tank Volume is calculated by multiplying the area of the base (length × width) by the height. This tells you exactly how much three-dimensional space is inside the tank.
1. Formula
- V = Volume
- l = Length of the tank (Usually the longest side of the base).
- w = Width of the tank (The shorter side of the base).
- h = Height of the tank (The vertical dimension extending upward).
Visual Representation of 3D Rectangular Tank
Volume = 24.00
2. Step-by-Step Calculation
Calculate the volume for a tank with a length of 10 cm, width of 5 cm, and height of 8 cm:
- Identify Dimensions: l = 10, w = 5, h = 8
- Calculate Base Area: 10 × 5 = 50
- Multiply by Height: 50 × 8 = 400
- Final Result: 400 cm3
3. Important Tips
- The Base Concept: Volume is always Base Area × Height. For a rectangle, the base area is just length × width.
- Check Your Units: If your length is in feet but your height is in inches, you must convert them to the same unit before multiplying.
- Capacity vs. Volume: Volume tells you the space inside; Capacity usually refers to how much liquid (Liters or Gallons) that volume can hold.
4. Applied Example: The Backyard Aquarium
A hobbyist is building a custom glass fish tank. The tank measures **1.2 meters long**, **0.5 meters wide**, and **0.6 meters high**. What is the total volume of water the tank can hold?
1. Identify the Knowns:
Length (l) = 1.2 m
Width (w) = 0.5 m
Height (h) = 0.6 m
2. Apply the Standard Formula:
V = l × w × h
3. Step-by-Step Calculation:
- • Calculate Base Area: 1.2 × 0.5 = 0.6
- • Multiply by Height: 0.6 × 0.6 = 0.36
Total Tank Volume:
Volume = 0.36 m3
Volume Formula Table
| Shape | Volume Formula |
|---|---|
| Sphere | 4/3 Ï€ r³ |
| Cylinder | Ï€ r² h |
| Cone | 1/3 Ï€ r² h |
| Cube | a³ |
| Pyramid | 1/3 × Base Area × Height |
| Capsule | Ï€r²h + 4/3Ï€r³ |
| Ellipsoid | 4/3Ï€abc |
| Frustum | 1/3Ï€h(R² + Rr + r²) |
| Rectangular Tank | L × W × H |
Volume Unit Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m³ | 1000 Liters |
| 1 Liter | 1000 cm³ |
| 1 cm³ | 0.001 Liters |
| 1 ft³ | 28.3168 Liters |
| 1 in³ | 16.387 cm³ |
Frequently Asked Questions
What is the unit of volume? What does volume represent?
Volume is the measure of the amount of space an object occupies. The standard unit of volume in the International System (SI) is the cubic meter (m³). Other common units include liters (L), cubic centimeters (cm³), cubic inches (in³), and cubic feet (ft³).
What is the volume of a cube in cubic meters?
The volume of a cube is calculated using the formula: V = a³, where "a" is the length of one side. If the side is measured in meters, the result will be in cubic meters (m³).
What is the volume of a cube with 1/2 inch sides?
Using the cube formula V = a³: V = (1/2)³ = 1/8 cubic inches.
What is the formula of the volume of a cone and hemisphere?
The volume of a cone is:
V = (1/3)Ï€r²h
The volume of a hemisphere is:
V = (2/3)Ï€r³
How to find the volume of a cylinder in litres?
First calculate the volume using:
V = Ï€r²h
Then convert cubic units to liters:
1 cubic meter = 1000 liters OR 1 cm³ = 0.001 liters.
How to find the volume of a rectangular prism in calculus?
In calculus, the volume of a rectangular prism can be found using integration:
V = ∫∫∫ dV
For a prism with constant dimensions, this simplifies to:
V = length × width × height.
How do you derive the volume of a sphere (4/3Ï€r³)?
The sphere volume is derived using integral calculus, specifically a technique called the Disk Method. We imagine a semi-circle being rotated 360 degrees around the x-axis to create a solid 3D object.
1. Setting the Equation
A circle centered at the origin (0,0) is defined as:
x² + y² = r²
To find the radius of our individual "disks," we solve for y:
y = √(r² - x²)
2. The Disk Method Concept
Imagine slicing the sphere into an infinite number of infinitesimally thin circular disks from -r to r. The volume of each tiny disk is pi times the radius squared:
3. The Integration Process
By summing all these disks together using an integral, we find the total space occupied by the dimensional object:
When you solve this definite integral, the math simplifies as follows:
- • Antiderivative: π [ r²x - (x³ / 3) ] from -r to r
- • Plugging in limits: π [ (r³ - r³/3) - (-r³ + r³/3) ]
- • Final Calculation: π [ 2r³ - 2r³/3 ] = π [ 4r³/3 ]
Final Result: V = 4⁄3 π r3
This confirms that the volume is exactly four-thirds pi times the radius cubed.
Why is the term "volume" used in sound?
In sound, "volume" refers to the loudness or intensity of audio. Although it does not measure physical space, the term is used metaphorically to describe how strong or powerful a sound wave is.