Calculating The Surface Area Of Cylinders In Terms Of Pi

🔑 Key Concepts

• A cylinder has 2 congruent circular bases and 1 curved lateral surface.
• The surface area formula for a cylinder is \(SA = 2\pi r^2 + 2\pi rh\).
• The term \(2\pi r^2\) represents the area of both circular bases.
• The term \(2\pi rh\) represents the lateral area around the cylinder.
• When the answer is in terms of \(\pi\), do not multiply by \(3.14\). Leave \(\pi\) in the final answer.
• The final answer should be written in square units.

✏️ Worked Example

Cylinder with Radius 10 cm and Height 25 cm

Given:

Radius: \(r = 10\text{ cm}\)

Height: \(h = 25\text{ cm}\)

Step 1: Use the cylinder surface area formula.

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

Step 2: Substitute the values.

\(SA = 2\pi(10)^2 + 2\pi(10)(25)\)

Step 3: Simplify each part.

\(2\pi(10)^2 = 2\pi(100)=200\pi\)

\(2\pi(10)(25)=500\pi\)

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

\(SA = 200\pi + 500\pi\)

\(SA = 700\pi\)

Surface Area:

\(SA = 700\pi\text{ cm}^2\)

🧠 Math Vocabulary

• cylinder: a three-dimensional figure with two congruent circular bases and one curved surface.
• radius: the distance from the center of a circle to a point on the circle.
• height: the distance between the two circular bases of the cylinder.
• base area: the area of one circular base, found using \(A=\pi r^2\).
• lateral area: the area around the curved side of the cylinder, found using \(2\pi rh\).
• in terms of \(\pi\): an exact answer that keeps \(\pi\) in the final expression.
• square units: units used to measure area, such as \( \text{cm}^2 \), \( \text{in}^2 \), or \( \text{ft}^2 \).

💡 Main Idea

To find the surface area of a cylinder, add the area of the two circular bases and the lateral area around the cylinder. When the problem asks for the answer in terms of \(\pi\), simplify the coefficient and leave \(\pi\) in the final answer.

📚 What You Should Already Know

Students should know how to find the area of a circle using \(A=\pi r^2\), square a number, multiply whole numbers, combine like terms, and understand that surface area measures the total outside covering of a three-dimensional figure.

🚀 What Comes Next

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

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