Advanced Geometric Shapes and Formulas

This page delves deeper into the properties of cylinders and introduces cones and spheres, providing formulas for calculating their volumes and surface areas.

The cylinder is further explored, with its key components identified: axis, radius, and generatrix. The formula for the volume of a cylinder is given as V = πr²h, where r is the radius of the base and h is the height.

Definition: The generatrix of a cylinder is the line segment that, when rotated around the axis, generates the lateral surface of the cylinder.

The cone of revolution is introduced, highlighting its key features such as the apex, base, and generatrix.

Vocabulary: The generatrix of a cone is the line segment from the apex to a point on the circumference of the base.

Formulas for the cone are provided, including:

• Circumference of the base: 2πr
• Area of the circular base: πr²
• Volume: V = (1/3)πr²h

Highlight: The volume of a cone is one-third the volume of a cylinder with the same base and height.

The sphere is briefly mentioned, with formulas for its surface area and volume:

• Surface area of a sphere: 4πr²
• Volume of a sphere (also called a ball): (4/3)πr³

Example: To calculate the volume of a sphere with a radius of 5 cm, you would use the formula V = (4/3)π(5³), which equals approximately 523.6 cm³.

This page provides a comprehensive overview of these advanced geometric shapes, offering essential formulas for volume and surface area calculations, which are crucial for students studying geometry and mathematics.