Cylinders Study Notes

A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Think of a can of soup or a pipe.

Understanding Cylinders 📐

Key Components:

• Radius (r): The distance from the center of the circular base to any point on its edge.
• Diameter (d): The distance across the circular base through its center. \(d = 2r\)
• Height (h): The perpendicular distance between the two circular bases.
• Base Area: The area of one of the circular bases. The formula for the area of a circle is \(A = \pi r^2\).

Formulas for Cylinders:

We will focus on two main calculations for cylinders: Volume and Surface Area.

Volume of a Cylinder 📦

The volume of a cylinder is the amount of space it occupies. It is calculated by multiplying the area of the base by the height of the cylinder.

The formula for the volume (V) of a cylinder is: \[ V = (\text{Base Area}) \times h \] Since the base is a circle, we substitute the formula for the area of a circle: \[ V = \pi r^2 h \]

💡 Pro Tip: Make sure your radius and height are in the same units before calculating the volume.

Example:

A cylinder has a radius of 5 cm and a height of 10 cm. What is its volume?

• \(r = 5\) cm
• \(h = 10\) cm
• \(V = \pi (5 \text{ cm})^2 (10 \text{ cm})\)
• \(V = \pi (25 \text{ cm}^2) (10 \text{ cm})\)
• \(V = 250\pi \text{ cm}^3\)

If you need to approximate using \(\pi \approx 3.14\), then \(V \approx 250 \times 3.14 = 785 \text{ cm}^3\).

Surface Area of a Cylinder 🌐

The surface area of a cylinder is the total area of all its surfaces, including the top and bottom bases and the curved side.

The formula for the total surface area (SA) of a cylinder is: \[ SA = (\text{Area of Top Base}) + (\text{Area of Bottom Base}) + (\text{Area of Curved Surface}) \] \[ SA = \pi r^2 + \pi r^2 + 2\pi rh \] \[ SA = 2\pi r^2 + 2\pi rh \] This can also be factored as: \[ SA = 2\pi r (r + h) \]

📌 Key Takeaway: The term \(2\pi r^2\) represents the area of the two circular bases, and \(2\pi rh\) represents the area of the curved lateral surface.

Example:

A cylinder has a radius of 3 inches and a height of 7 inches. What is its total surface area?

• \(r = 3\) inches
• \(h = 7\) inches
• \(SA = 2\pi (3 \text{ inches})^2 + 2\pi (3 \text{ inches})(7 \text{ inches})\)
• \(SA = 2\pi (9 \text{ inches}^2) + 2\pi (21 \text{ inches}^2)\)
• \(SA = 18\pi \text{ inches}^2 + 42\pi \text{ inches}^2\)
• \(SA = 60\pi \text{ inches}^2\)

Using \(\pi \approx 3.14\), \(SA \approx 60 \times 3.14 = 188.4 \text{ inches}^2\).

Calculation	Formula	Description
Volume	\(V = \pi r^2 h\)	Space occupied by the cylinder.
Surface Area	\(SA = 2\pi r^2 + 2\pi rh\)	Total area of all surfaces.