The height and radius of the right circular cylinder below are given in meters. What is the volume, in cubic meters, of the cylinder? A. 30\(\pi\) B. 31\(\pi\) C. 150\(\pi\) D. 180\(\pi\) E. 900\(\pi\)

Geometry is a branch of mathematics that deals with different shapes and the properties associated with them. At its core, geometry allows us to understand spatial relationships and provides tools to measure and compare various dimensions. In the realm of geometry, objects such as points, lines, surfaces, and solids are studied both in the abstract sense and in their practical applications.

For instance, when considering 3-dimensional objects, one of the most straightforward figures to analyze is the cylinder, known for its use in everyday objects like cans and pipes. Geometry not only helps us appreciate the aesthetics of these shapes but also equips us with the knowledge to calculate important attributes, such as surface area and volume — critical for designing, manufacturing, and utilizing these objects effectively.

A circular cylinder is a 3-dimensional shape with two parallel circular bases connected by a curved surface. The axis of the cylinder is the straight line connecting the centers of the two bases. When this axis is perpendicular to the bases, the cylinder is termed a 'right circular cylinder.'

Key Attributes

• Axis: the straight line between the centers of the two bases.
• Height (h): the distance between the bases, measured along the axis.
• Radius (r): the distance from the center to the edge of the circular base.
• Base Area: the area of one of the circular bases, calculated as \( A_{base} = \)\(\pi r^2 \)\).

These characteristics are essential in determining properties like the cylinder's volume, and they play a major role in practical applications such as calculating the capacity of tanks or the displacement in engines.

For right circular cylinders, the volume is a measurement indicating how much space the shape occupies. It's especially useful when you need to calculate how much liquid a cylindrical container can hold or the amount of air in a cylinder.

Mathematically, the volume of a right circular cylinder is given by the formula:\[ V = \( \pi r^2 h \) \]. In this formula, \( r \) is the radius of the base, and \( h \) is the height of the cylinder. The \( \pi \) symbol represents a constant approximately equal to 3.14159. This constant is the ratio of the circumference of a circle to its diameter and is universal for all circles. By squaring the radius, we find the area of the base (\( A_{base} = \pi r^2 \) ), and multiplying by the height gives us the cylinder's volume.

When calculating volume, careful attention must be paid to the units used for radius and height; the volume's units will be in cubic terms. For example, if radius and height are measured in meters, the resulting volume will be in cubic meters (m³). It's crucial to use consistent units to ensure the accuracy of the calculation.