Volume Calculator

Understanding Volume

Volume is the quantification of the three-dimensional space a substance occupies. The SI unit for volume is the cubic meter (m³). Typically, the volume of a container refers to its capacity—the amount of fluid it can hold—rather than the space the container itself occupies. Volumes of many shapes can be calculated using well-defined formulas. Complex shapes can often be broken down into simpler ones, and the sum of their volumes gives the total volume. For even more complicated shapes, integral calculus or numerical methods may be used. Below is a comprehensive guide to calculating the volume of some common shapes:

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Sphere

A sphere is a perfectly round geometrical object in three-dimensional space. Every point on the surface of a sphere is equidistant from its center.

Volume Formula:

$$V = \frac{4}{3}\pi r^3$$

• $V$    = Volume
• $r$    = Radius of the sphere

Example:

Claire wants to fill a spherical water balloon with a radius of 0.15 ft. The volume $V$ is: $$V = \frac{4}{3}\pi (0.15)^3 \approx 0.141 \text{ ft}^3$$

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Cone

A cone tapers smoothly from a flat base to a point called the apex.

Volume Formula:

$$V = \frac{1}{3}\pi r^2 h$$

• $V$    = Volume
• $r$  = Radius of the base
• $h$    = Height of the cone

Example:

Bea wants to know if a waffle cone (radius 1.5 in, height 5 in) holds at least 15% more ice cream than a sugar cone. The volume is: $$V = \frac{1}{3}\pi (1.5)^2 (5) \approx 11.78 \text{ in}^3$$

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Cube

A cube has six equal square faces.

Volume Formula:

$$V = a^3$$

• $V$    = Volume
• $a$  = Edge length of the cube

Example:

Bob wants to fill his cubic suitcase (edge length 2 ft) with soil: $$V = (2)^3 = 8 \text{ ft}^3$$

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Cylinder

A cylinder has circular bases connected by a curved surface.

Volume Formula:

$$V = \pi r^2 h$$

• $V$  = Volume
• $r$  = Radius of the base
• $h$  = Height of the cylinder

Example:

Caelum wants to know how much sand his barrels can hold (radius 3 ft, height 4 ft): $$V = \pi (3)^2 (4) \approx 113.10 \text{ ft}^3$$

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Rectangular Tank

A rectangular tank is a box-shaped object.

Volume Formula:

$$V = l \times w \times h$$

• $V$  = Volume
• $l$  = Length
• $w$    = Width
• $h$  = Height

Example:

Darby wants to fill her backpack (4 ft x 3 ft x 2 ft) with cake: $$V = 4 \times 3 \times 2 = 24 \text{ ft}^3$$

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Capsule

A capsule consists of a cylinder with hemispherical ends.

Volume Formula:

$$V = \pi r^2 \left( h + \frac{4}{3}r \right)$$

• $V$  = Volume
• $r$  = Radius
• $h$    = Height of the cylindrical part

Example:

Joe wants to know the volume of his capsule-shaped time capsule (radius 1.5 ft, cylinder height 3 ft): $$V = \pi (1.5)^2 \left( 3 + \frac{4}{3}(1.5) \right) \approx 35.34 \text{ ft}^3$$

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Spherical Cap

A spherical cap is a portion of a sphere cut off by a plane.

Volume Formula:

$$V = \frac{1}{3}\pi h^2 (3R – h)$$

• $V$  = Volume
• $h$    = Height of the cap
• $R$    = Radius of the sphere

Example:

Jack calculates the volume of a spherical cap from a golf ball (radius 1.68 in, cap height 0.3 in): $$V = \frac{1}{3}\pi (0.3)^2 (3 \times 1.68 – 0.3) \approx 0.447 \text{ in}^3$$

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Conical Frustum

A conical frustum is a cone with the top cut off parallel to the base.

Volume Formula:

$$V = \frac{1}{3}\pi h \left( r_1^2 + r_1 r_2 + r_2^2 \right)$$

• $V$  = Volume
• $h$  = Height of the frustum
• $r_1$  = Radius of the lower base
• $r_2$    = Radius of the upper base

Example:

Bea calculates the volume of her ice cream cone after her brother bites off the tip (lower radius 1.5 in, upper radius 0.2 in, height 4 in): $$V = \frac{1}{3}\pi (4) \left( (1.5)^2 + 1.5 \times 0.2 + (0.2)^2 \right) \approx 10.85 \text{ in}^3$$

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Ellipsoid

An ellipsoid is a stretched sphere with three principal axes.

Volume Formula:

$$V = \frac{4}{3}\pi a b c$$

• $V$  = Volume
• $a, b, c$  = Semi-axes lengths

Example:

Xabat calculates the volume of meat he can fit into an ellipsoid-shaped bun (axes 1.5 in, 2 in, 5 in): $$V = \frac{4}{3}\pi (1.5)(2)(5) \approx 62.83 \text{ in}^3$$

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Square Pyramid

A square pyramid has a square base and triangular sides.

Volume Formula:

$$V = \frac{1}{3} a^2 h$$

• $V$  = Volume
• $a$  = Base edge length
• $h$  = Height

Example:

Wan calculates the volume of a mud pyramid (base edge 5 ft, height 12 ft): $$V = \frac{1}{3} (5)^2 (12) = 100 \text{ ft}^3$$

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Tube

A tube is a hollow cylinder.

Volume Formula:

$$V = \pi h \left( r_{\text{outer}}^2 – r_{\text{inner}}^2 \right)$$

• $V$  = Volume
• $h$  = Height or length of the tube
• $r_{\text{outer}}$  = Outer radius
• $r_{\text{inner}}$  = Inner radius

Example:

Beulah calculates the volume of concrete needed for a pipe (outer diameter 3 ft, inner diameter 2.5 ft, length 10 ft): Convert diameters to radii: $$r_{\text{outer}} = \frac{3}{2} = 1.5 \text{ ft}, \quad r_{\text{inner}} = \frac{2.5}{2} = 1.25 \text{ ft}$$ Calculate volume: $$V = \pi (10) \left( (1.5)^2 – (1.25)^2 \right) \approx 21.6 \text{ ft}^3$$

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Common Volume Units

Unit	Cubic Meters (m³)	Milliliters (ml)
Milliliter (cc)	0.000001	1
Cubic Inch	0.00001639	16.39
Pint	0.000473	473
Quart	0.000946	946
Liter	0.001	1,000
Gallon	0.003785	3,785
Cubic Foot	0.028317	28,317
Cubic Yard	0.764555	764,555
Cubic Meter	1	1,000,000
Cubic Kilometer	1,000,000,000	$10^{15}$1015

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Conclusion

Understanding how to calculate the volume of various shapes is essential in fields like engineering, architecture, and everyday life situations. Use the volume calculator above to compute volumes by entering the required dimensions. Note: Always ensure that all measurements are in the same units before performing calculations.