Volume is the amount of three-dimensional space enclosed by a closed surface. It quantifies how much space an object or substance occupies. Understanding volume is crucial in various fields, including physics, engineering, and everyday life.

The method for calculating volume depends on the shape of the object. Each shape has its own specific formula. Let's explore the volume formulas for different shapes:

\[ V = \frac{4}{3}\pi r^3 \]

Where:

- \(V\) is the volume
- \(r\) is the radius of the sphere

Calculation steps:

- Measure the radius of the sphere
- Cube the radius (multiply it by itself twice)
- Multiply the result by \(\frac{4}{3}\pi\) (approximately 4.1887)

Example: For a sphere with radius 5 cm

\[ V = \frac{4}{3}\pi (5\text{ cm})^3 = \frac{4}{3}\pi (125\text{ cm}^3) \approx 523.6\text{ cm}^3 \]

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

Where:

- \(V\) is the volume
- \(r\) is the radius of the base
- \(h\) is the height of the cone

Calculation steps:

- Measure the radius of the base and the height of the cone
- Square the radius
- Multiply the squared radius by the height
- Multiply the result by \(\frac{1}{3}\pi\) (approximately 1.0472)

Example: For a cone with base radius 3 cm and height 4 cm

\[ V = \frac{1}{3}\pi (3\text{ cm})^2 (4\text{ cm}) = \frac{1}{3}\pi (36\text{ cm}^3) \approx 37.7\text{ cm}^3 \]

\[ V = \pi r^2 h \]

Where:

- \(V\) is the volume
- \(r\) is the radius of the base
- \(h\) is the height of the cylinder

Calculation steps:

- Measure the radius of the base and the height of the cylinder
- Square the radius
- Multiply the squared radius by the height
- Multiply the result by \(\pi\) (approximately 3.14159)

Example: For a cylinder with base radius 5 cm and height 10 cm

\[ V = \pi (5\text{ cm})^2 (10\text{ cm}) = \pi (250\text{ cm}^3) \approx 785.4\text{ cm}^3 \]

\[ V = l \times w \times h \]

Where:

- \(V\) is the volume
- \(l\) is the length
- \(w\) is the width
- \(h\) is the height

Calculation steps:

- Measure the length, width, and height of the rectangular tank
- Multiply these three measurements together

Example: For a rectangular tank with length 5 cm, width 3 cm, and height 4 cm

\[ V = 5\text{ cm} \times 3\text{ cm} \times 4\text{ cm} = 60\text{ cm}^3 \]

\[ V = \frac{4}{3}\pi r^3 + \pi r^2 h \]

Where:

- \(V\) is the volume
- \(r\) is the radius of the cylindrical part and the spherical ends
- \(h\) is the height of the cylindrical part

Calculation steps:

- Calculate the volume of a sphere with the given radius (\(\frac{4}{3}\pi r^3\))
- Calculate the volume of a cylinder with the given radius and height (\(\pi r^2 h\))
- Add these two volumes together

Example: For a capsule with radius 2 cm and cylindrical height 5 cm

\[ V = \frac{4}{3}\pi (2\text{ cm})^3 + \pi (2\text{ cm})^2 (5\text{ cm}) \approx 33.5\text{ cm}^3 + 62.8\text{ cm}^3 = 96.3\text{ cm}^3 \]

Cap volume refers to the volume of a portion of a sphere cut off by a plane. It is a common concept in geometry and has practical applications in various fields, including engineering and physics.

\[ V = \frac{1}{3}\pi h^2(3R - h) \]

Where:

- \(V\) is the volume of the cap
- \(h\) is the height of the cap
- \(R\) is the radius of the sphere
- \(r\) is the radius of the base of the cap

- Determine the values of \(r\), \(R\), and \(h\)
- Calculate \(h^2\)
- Calculate \(3R - h\)
- Multiply the results from steps 2 and 3
- Multiply the result by \(\frac{1}{3}\pi\) (approximately 1.0472)

Let's calculate the volume of a spherical cap with a base radius (\(r\)) of 4 cm, cut from a sphere with radius (\(R\)) of 5 cm.

First, we need to calculate the height (\(h\)) of the cap using the Pythagorean theorem:

\[ h = R - \sqrt{R^2 - r^2} = 5 - \sqrt{5^2 - 4^2} \approx 1.66 \text{ cm} \]

Now we can apply the volume formula:

\[ \begin{align*} V &= \frac{1}{3}\pi h^2(3R - h) \\ &= \frac{1}{3}\pi (1.66)^2(3(5) - 1.66) \\ &\approx 1.0472 \times 2.7556 \times 13.34 \\ &\approx 38.48 \text{ cm}^3 \end{align*} \]

Therefore, the volume of the spherical cap is approximately 38.48 cubic centimeters.

\[ V = \frac{1}{3}\pi h(R^2 + r^2 + Rr) \]

Where:

- \(V\) is the volume
- \(h\) is the height of the frustum
- \(R\) is the radius of the larger base
- \(r\) is the radius of the smaller base

Calculation steps:

- Measure the height of the frustum and the radii of both bases
- Calculate \(R^2\), \(r^2\), and \(Rr\)
- Add these three results
- Multiply by the height and \(\frac{1}{3}\pi\) (approximately 1.0472)

Example: For a conical frustum with height 10 cm, larger base radius 5 cm, and smaller base radius 3 cm

\[ V = \frac{1}{3}\pi (10\text{ cm})((5\text{ cm})^2 + (3\text{ cm})^2 + (5\text{ cm})(3\text{ cm})) \approx 366.5\text{ cm}^3 \]

\[ V = \frac{4}{3}\pi abc \]

Where:

- \(V\) is the volume
- \(a\), \(b\), and \(c\) are the semi-axes lengths

Calculation steps:

- Measure the lengths of the three semi-axes
- Multiply these three lengths together
- Multiply the result by \(\frac{4}{3}\pi\) (approximately 4.1887)

Example: For an ellipsoid with semi-axes lengths 3 cm, 4 cm, and 5 cm

\[ V = \frac{4}{3}\pi (3\text{ cm})(4\text{ cm})(5\text{ cm}) = \frac{4}{3}\pi (60\text{ cm}^3) \approx 251.3\text{ cm}^3 \]

\[ V = \frac{1}{3}l^2h \]

Where:

- \(V\) is the volume
- \(l\) is the length of one side of the base
- \(h\) is the height of the pyramid

Calculation steps:

- Measure the length of one side of the base and the height of the pyramid
- Square the base side length
- Multiply the squared base side length by the height
- Multiply the result by \(\frac{1}{3}\) (approximately 0.3333)

Example: For a square pyramid with base side length 6 cm and height 8 cm

\[ V = \frac{1}{3}(6\text{ cm})^2(8\text{ cm}) = \frac{1}{3}(288\text{ cm}^3) = 96\text{ cm}^3 \]

Understanding these volume formulas and how to apply them is crucial in many fields, including engineering, architecture, and manufacturing. They allow us to calculate the capacity of containers, the amount of material needed for construction, or even the buoyancy of objects in fluids.

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