In this article, we will learn how to calculate the volume and area of the right cylinder.

Right cylinders are commonly found in everyday objects such as cans and tubes. They are three-dimensional objects with a circular base and straight sides that are perpendicular to the base. Calculating various properties of right cylinders, such as volume and surface area, is important in many fields, including engineering and manufacturing.

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## Calculating the Volume of a Right Cylinder

The volume of a right cylinder can be calculated using the formula:

V = \pi r^2h

where V is the volume, r is the radius, and \$h[/latex] is the height.

## Determining the Surface Area of a Right Cylinder

The surface area of a right cylinder can be broken down into two parts: the lateral surface area and the base area. The lateral surface area of a right cylinder is given by the formula:

L = 2\pi rh

where L is the lateral surface area. The base area of a right cylinder is simply the area of the circular base, which is given by:

B = \pi r^2

The total surface area of a right cylinder can be calculated by adding the lateral surface area and the base area:

A = L + 2B = 2\pi rh + 2\pi r^2

where A is the total surface area.

## Finding the Height of a Right Cylinder

The height of a right cylinder can be found using the formula:

h = \frac{V}{\pi r^2}

where V is the volume and r is the radius.

## Computing the Radius of a Right Cylinder

The radius of a right cylinder can be computed using the formula:

r = \sqrt{\frac{V}{\pi h}}

where V is the volume and h is the height.

## Determining the Diameter of a Right Cylinder

The diameter of a right cylinder is simply twice the radius:

d = 2r

where d is the diameter and r is the radius.

## Converting Units in Right Cylinder Calculations

When performing calculations involving right cylinders, it is important to make sure that all measurements are in the same units. If the height is given in inches and the radius is given in centimeters, for example, one of the measurements should be converted to match the other before performing calculations.

## Examples of Right Cylinder Calculations

Example 1: A can of soup has a radius of 3 cm and a height of 10 cm. What is the volume of the soup in the can?

Solution: Using the formula for the volume of a right cylinder, we have:

V = \pi r^2h = \pi (3\text{ cm})^2(10\text{ cm}) \approx 282.74\text{ cm}^3

Therefore, the volume of soup in the can is approximately 282.74 cubic centimeters.

Example 2: A tube of toothpaste has a radius of 2 cm and a height of 15 cm. What is the total surface area of the tube?

Solution: Using the formulas for the lateral surface area and base area of a right cylinder, we have:

L = 2\pi rh = 2\pi (2\text{ cm})(15\text{ cm}) = 60\pi\text{ cm}^2

and

B = \pi r^2 = \pi (2\text{ cm})^2 = 4\pi\text{ cm}^2

Therefore, the total surface area of the tube is:

{ cm}^2 + 8\pi\text{ cm}^2 \approx 87.96\text{ cm}^2

Example 3: Find the height of a cylinder with a volume of 500 cm³ and a radius of 5 cm.

Solution: Using the formula for the height of a right cylinder, we have:

h = \frac{V}{\pi r^2} = \frac{500\text{ cm}^3}{\pi (5\text{ cm})^2} \approx 6.37\text{ cm}

Therefore, the height of the cylinder is approximately 6.37 centimeters.

Example 4: Find the radius of a cylinder with a volume of 1000 m³ and a height of 20 m.

Solution: Using the formula for the radius of a right cylinder, we have:

r = \sqrt{\frac{V}{\pi h}} = \sqrt{\frac{1000\text{ m}^3}{\pi (20\text{ m})}} \approx 2.52\text{ m}

Therefore, the radius of the cylinder is approximately 2.52 meters.

Example 5: Find the diameter of a cylinder with a radius of 6 ft.

Solution: Using the formula for the diameter of a right cylinder, we have:

d = 2r = 2(6\text{ ft}) = 12\text{ ft}

Therefore, the diameter of the cylinder is 12 feet.

These examples demonstrate the various calculations that can be performed using the formulas for a right cylinder. It is important to carefully read the problem statement and identify which formula is required to solve the problem. Additionally, converting units to ensure consistency is crucial in accurate calculations.

## Converting Units in Right Cylinder Calculations

When working with right cylinders, it is important to ensure that all measurements are in the same units before performing any calculations. For example, if the radius is given in inches and the height is given in feet, the units need to be converted to either inches or feet before using any of the formulas.

To convert units, we can use conversion factors. A conversion factor is a ratio of two equivalent measurements in different units. For example, 1 meter is equivalent to 3.28 feet, so we can use the conversion factor 1 m = 3.28 ft to convert from meters to feet or vice versa.

Example: Convert a cylinder with a radius of 4 inches and a height of 10 feet to all units in inches.

Solution: First, we need to convert the height from feet to inches. Using the conversion factor 1 ft = 12 in, we have:

10\text{ ft} \cdot \frac{12\text{ in}}{1\text{ ft}} = 120\text{ in}

Now, we have the radius in inches and the height in inches, so we can use the formulas for the volume, surface area, etc.

## Examples of Right Cylinder Calculations

Example 1: A cylindrical water tank has a radius of 5 feet and a height of 10 feet. Find the volume of water the tank can hold.

Solution: Using the formula for the volume of a right cylinder, we have:

V = \pi r^2 h = \pi (5\text{ ft})^2 (10\text{ ft}) = 250\pi\text{ ft}^3 \approx 785.4\text{ ft}^3

Therefore, the tank can hold approximately 785.4 cubic feet of water.

Example 2: A cylindrical can has a height of 8 cm and a diameter of 6 cm. Find the surface area of the can.

Solution: Using the formula for the surface area of a right cylinder, we have:

A = 2\pi rh + 2\pi r^2 = 2\pi (3\text{ cm}) (8\text{ cm}) + 2\pi (3\text{ cm})^2 \approx 113.1\text{ cm}^2

Therefore, the surface area of the can is approximately 113.1 square centimeters.

These examples illustrate the practical applications of right cylinder calculations in real-world scenarios. Understanding how to use the formulas for volume, surface area, height, radius, and diameter can be useful in fields such as engineering, architecture, and construction.

## FAQ

### What is a cylinder?

A cylinder is a geometric shape that can be defined as a solid figure with two parallel bases connected by a curved lateral surface.

### What is the height of a cylinder?

The height of a cylinder is the distance between the two bases of the cylinder.

### What is the diagonal of a cylinder?

The diagonal of a cylinder is the longest distance you can find inside the cylinder. In other words, it is the longest distance between the two bases of the cylinder.