Surface Area Of A Cylinder

🔑 Key Concepts

• A cylinder net is made of 2 congruent circles and 1 rectangle.
• The two circles represent the top and bottom bases of the cylinder.
• The rectangle represents the curved lateral surface when it is unwrapped.
• The width of the rectangle is the circumference of the circular base, \(2\pi r\).
• The height of the rectangle is the height of the cylinder.
• To find surface area, add the area of the two circles and the area of the rectangle.

✏️ Worked Example

Cylinder with Radius 2 cm and Height 5 cm

Given:

Radius: \(r = 2\text{ cm}\)

Height: \(h = 5\text{ cm}\)

Step 1: Use the cylinder surface area formula.

\(SA = 2\pi r^2 + 2\pi rh\)

Step 2: Substitute the values.

\(SA = 2\pi(2)^2 + 2\pi(2)(5)\)

Step 3: Simplify each part.

\(2\pi(2)^2 = 2\pi(4)=8\pi\)

\(2\pi(2)(5)=20\pi\)

Step 4: Add the \(\pi\)-terms.

\(SA = 8\pi + 20\pi\)

\(SA = 28\pi\)

Exact Surface Area:

\(SA = 28\pi\text{ cm}^2\)

Approximate Surface Area:

\(28\pi \approx 28(3.14)\)

\(28(3.14)=87.92\)

\(SA \approx 87.92\text{ cm}^2\)

🧠 Math Vocabulary

• cylinder: a three-dimensional figure with two congruent circular bases and one curved surface.
• net: a flat pattern that represents all faces or surfaces of a three-dimensional figure.
• radius: the distance from the center of a circular base to the edge of the circle.
• height: the distance between the two circular bases of a cylinder.
• circumference: the distance around a circle, found using \(C=2\pi r\).
• lateral area: the area of the curved side of a cylinder, represented by the rectangle in the net.
• base area: the area of one circular base, found using \(A=\pi r^2\).
• surface area: the total area of all outside surfaces of a three-dimensional figure.

💡 Main Idea

A cylinder net helps explain why the surface area formula works. The two circles give the area of the bases, and the rectangle gives the lateral area. Since the rectangle wraps around the cylinder, its length is the circumference of the circular base and its height is the height of the cylinder.

📚 What You Should Already Know

Students should know how to find the area of a circle using \(A=\pi r^2\), how to find circumference using \(C=2\pi r\), how to multiply with \(\pi\), and how to add areas together to find total surface area.

🚀 What Comes Next

After connecting cylinder nets to surface area, students can compare exact answers in terms of \(\pi\) with decimal approximations. They can also apply cylinder surface area to real-world problems involving labels, cans, containers, and packaging.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning management system.