Question

A cylinder has a base diameter of 14 inches and a height of 18 inches. What is the volume of the cylinder?

Short answer

The volume of the cylinder is approximately 2769.48 cubic inches.

Step-by-step solution

Understand the Formula for Volume of a Cylinder

To find the volume of a cylinder, we use the formula: \[ V = \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.

Identify Given Values

The diameter of the cylinder's base is 14 inches. The radius \( r \) is half of the diameter, so \( r = \frac{14}{2} = 7 \) inches. The height \( h \) of the cylinder is given as 18 inches.

Substitute Values into the Formula

Substitute the known values into the volume formula: \[ V = \pi (7)^2 (18) \] This simplifies to: \[ V = \pi \cdot 49 \cdot 18 \]

Calculate the Volume

Calculate the volume by first performing the multiplication inside the parentheses: \[ 49 \times 18 = 882 \] Now multiply by \( \pi \): \[ V = 882\pi \] If we approximate \( \pi \) as 3.14, then: \[ V \approx 882 \times 3.14 = 2769.48 \]

Review the Units

Remember that volume is measured in cubic units, so the volume should be reported in cubic inches. Therefore, the volume is approximately 2769.48 cubic inches.

Key concepts

Understanding Geometry

Geometry is the branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. It plays a crucial role in understanding how shapes and their dimensions relate to each other. When we talk about a cylinder in geometry, we are dealing with a three-dimensional shape.
In geometry, the study of shapes is not only about identifying them but also about understanding their properties and measurements.

• Properties include aspects such as angles, faces, edges, and vertices.
• Measurements involve calculating areas, volumes, and surface areas.

Geometry is foundational as it helps us visualize and comprehend the space around us, enabling us to solve real-world problems involving shapes like cylinders.

What is a Cylinder?

A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Imagine a soda can—in geometry, that's a perfect example of a cylinder.
Some features of a cylinder include:

• A circular base and top, equal in size.
• A height, which is the distance between the two bases.
• A constant radius from the center of the base to its perimeter.

Cylinders are prevalent in everyday objects, from containers to mechanical parts, and understanding their volume is essential for various applications.

Volume Formula for a Cylinder

Calculating the volume of a cylinder involves measuring the space it occupies. The formula to find this is: \[ V = \pi r^2 h \]where:

• \( V \) is the volume, which tells us how much space is inside the cylinder.
• \( r \) is the radius of the circular base, which is half the diameter.
• \( h \) is the height of the cylinder.

Applying this formula is like stacking circles (the base) upwards for the given height of the cylinder, to find out how many cubic units the cylinder can hold. This formula showcases the interconnectedness of different geometric properties and their role in spatial calculations.

Mathematics Education and its Importance

Learning mathematics, including geometry and volume calculations, provides essential problem-solving tools. Understanding the volume of a cylinder is a building block in math education. It enhances logical thinking and helps students apply mathematical concepts to real-life situations.
Education systems emphasize such concepts to help students:

• Develop critical thinking skills.
• Understand practical applications of math in everyday life, like measuring objects or liquids.
• Build a foundation for more advanced studies in science, engineering, and technology.

Mathematics education prepares students for careers dependent on understanding these principles, fostering skills necessary for innovation and technological advancement.