10. Surface Area of a Cylinder

Session 10 Learning Plan

Learning objective: Calculate full and partial surface area of cylinders.

Key vocabulary: circular base, curved surface, circumference, lateral area

Success criterion: Use a cylinder net to select \(2\pi r^2\), \(2\pi rh\), or both.

Learn and Practice

A cylinder's surface has three parts: the top circle, the bottom circle, and the side that wraps around. Unrolling the side gives a rectangle whose width is the circumference (\(2\pi r\)) and whose height is \(h\).

Putting the pieces together:
\[ SA = \underbrace{2\pi r^2}_{\text{two circles}} + \underbrace{2\pi r h}_{\text{the wrap}}. \]

Worked example: a cylinder with radius 3 cm and height 10 cm has
\[ SA = 2(3.14)(3^2) + 2(3.14)(3)(10) = 56.52 + 188.4 = 244.92 \text{ cm}^2. \]

A label on a can (with no top or bottom) is just the wrap-around part, \(2\pi r h\).

Case study - making a closed tin can: the metal needed is the full surface area, both circles plus the side.

Common mistake: including only the side or only the circles. A closed cylinder needs both: two circles and the wrap-around rectangle.

Unroll the Curved Surface

Cut the curved side of a cylinder vertically and lay it flat. It becomes a rectangle. The rectangle's height is the cylinder height \(h\), and its width is the distance around the circular base, \(2\pi r\). Therefore its area is
\[ (2\pi r)h=2\pi rh. \]
Adding two circular ends gives the full closed-cylinder formula \(SA=2\pi r^2+2\pi rh\).

For radius \(4\text{ cm}\) and height \(7\text{ cm}\):
\[ \begin{aligned} \text{two ends}&=2\pi(4)^2=32\pi,\\ \text{curved side}&=2\pi(4)(7)=56\pi,\\ \text{total}&=88\pi\approx276.5\text{ cm}^2. \end{aligned} \]

Match the Surface to the Situation

A paper label covers only the curved side, so use \(2\pi rh\). An open cup has one circular base plus the curved side, so use \(\pi r^2+2\pi rh\). A sealed can has both bases and the side. Rather than memorizing three separate formulas, draw or list the included pieces.

Avoid a Volume Mix-Up

Both cylinder formulas contain \(\pi\), \(r\), and \(h\), but their structures reveal their meaning. Volume has one base area times height: \(\pi r^2h\), producing cubic units. Surface area adds flat regions: \(2\pi r^2+2\pi rh\), producing square units. Check the operation and the unit together.

GED® Coaching: A Reliable Five-Pass Method

Use the same disciplined process on every three-dimensional measurement problem.

1. Name the target. Write volume or surface area before calculating. Words such as hold, fill, capacity, and inside usually signal volume. Words such as cover, wrap, paint, label, and outside usually signal surface area.
2. Identify the solid and the given measurements. Mark length, width, height, radius, diameter, base area, or slant height. Never assume a drawing is to scale. If a diameter is given but the formula uses radius, write \(r=d/2\) before substituting.
3. Write the formula in symbols. Use the GED® formula sheet, but choose the formula from the meaning of the problem. Then substitute values with units. This makes a missing square, cube, or one-third factor much easier to spot.
4. Calculate in visible stages. Evaluate exponents first, then multiplication and division. Keep \(\pi\) in the calculator until the final step unless the question directs you to use \(3.14\). Round only once, at the end.
5. Verify meaning, size, and unit. Volume must use cubic units; surface area must use square units. Estimate with friendly numbers and ask whether the answer fits the situation. A cone matching a cylinder must have one-third of its volume, and a closed container must include every exposed face.

When an answer is wrong, classify the cause rather than merely recording the score: target error, solid-identification error, formula error, radius/diameter error, exponent error, unit error, calculation error, or rounding error. Rewrite the solution from the first incorrect decision. This correction routine turns each missed item into a reusable GED® strategy.

Session Practice Routine

1. State whether the target is interior volume or exterior surface area.
2. Name the solid and mark only the required measurements.
3. Convert dimensions to compatible units before using a formula.
4. Write the formula, substitute visibly, and evaluate in stages.
5. Attach square or cubic units and verify the size by estimation.
6. Complete the interactive self-check, then explain any correction aloud.

Exit Reflection

Without looking back, sketch the session's solid or net, write the key formula
from meaning, name one likely error, and describe the check that would catch it.