Cylinder Volume Calculation

• A cylinder is a three-dimensional geometric shape with straight parallel sides and circular or oval ends.
• Common examples include drinking cans or paper towel rolls.
• The volume of a cylinder is the amount of space it occupies.

Key Formula

• The volume (V) of a cylinder is calculated using the formula: V = πr²h, where 'r' is the radius of the circular base, and 'h' is the height of the cylinder.
• π (Pi) is a mathematical constant approximately equal to 3.14159.
• The radius (r) is the distance from the center of the circle to any point on its circumference.
• The height (h) is the perpendicular distance between the two circular bases.
• Both radius and height must be in the same units.

Formula Derivation

• The volume of a cylinder can be understood as the area of its base multiplied by its height.
• The base of a cylinder is a circle with area A = πr².
• The volume is then the area of this circular base extended along the height 'h', thus V = πr²h.
• This formula applies to right cylinders, where the height is perpendicular to the base.

Alternative Formula using Diameter

• If the diameter (d) of the cylinder is known instead of the radius, the formula can be expressed as: V = π(d/2)²h.
• The radius is half of the diameter (r = d/2).
• Simplifying, this becomes V = π(d²/4)h.

Oblique Cylinders

• For oblique cylinders (where the side is not perpendicular to the base), the formula V = πr²h still applies.
• 'h' must be the perpendicular height between the bases.

Units of Measurement

• The units of volume are cubic units (e.g., cm³, m³, in³).
• This depends on the units used for the radius and height
• If the radius and height are measured in centimeters (cm), the volume is in cubic centimeters (cm³).
• If the radius and height are measured in meters (m), the volume is in cubic meters (m³).

Practical Applications

• Calculating the amount of liquid a cylindrical tank can hold.
• Determining the material needed to construct a cylindrical object.
• Useful in engineering.

Example Calculation

• Consider a cylinder with a radius of 5 cm and a height of 10 cm.
• Using the formula V = πr²h, the volume is V = π * (5 cm)² * 10 cm = π * 25 cm² * 10 cm = 250π cm³.
• Approximating π as 3.14159, the volume is approximately 250 * 3.14159 cm³ ≈ 785.4 cm³.

Using the Formula

• Identify the radius (r) and height (h) of the cylinder.
• Ensure that the radius and height are in the same units.
• Substitute the values of 'r' and 'h' into the formula V = πr²h.
• Calculate the volume, using a calculator for the value of π.
• Express the volume in cubic units.

Volume vs Surface Area

• Volume measures the space inside the cylinder.
• Surface area measures the total area of the outer surface of the cylinder.
• Surface area includes the areas of the top and bottom circles, as well as the curved side.
• The formula for the surface area (SA) of a cylinder is SA = 2πr² + 2πrh.

Key Properties of Cylinders

• A cylinder has two circular bases that are parallel and congruent.
• The axis of a cylinder is the line segment connecting the centers of the two bases.
• In a right cylinder, the axis is perpendicular to the bases.
• In an oblique cylinder, the axis is not perpendicular to the bases.

Cavalieri's Principle

• Cavalieri's Principle states that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
• This principle can be used to show that the volume of an oblique cylinder is the same as the volume of a right cylinder with the same base area and height.

Real-World Examples

• Food cans.
• Water tanks.
• Pipes.
• Rollers.
• Some types of packaging.