MATH 130 · Unit 2

Cylinders: Volume and Surface Area

Wednesday, October 21, 2026 · 30-minute solids lesson after the 15-minute DMS-only Quiz 8

30 minutes · Offline capable

Today’s sequence

  1. 1

    Quiz 8 · 15 minutes · decimal degrees → DMS only

  2. 2

    Lesson · distinguish capacity from exposed surface

  3. 3

    Model and practice closed-cylinder volume and total surface area

  4. 4

    Extend to material volume in a hollow cylinder

Activate · 0 minSource: Authored

Capacity or covering?

Predict first

A sealed can holds a drink and is painted over every outside surface, including both circular ends. Which measurement describes the drink it can hold? Which describes the paint needed?

Commit to a response before discussion.

Activate · 3 minSource: Authored

Cylinder volume and surface-area formulas

Formula focus
  • Base area and circumference: $A_{\mathrm{base}}=\pi r^2$, $C=2\pi r$.
  • Volume (capacity): $V=A_{\mathrm{base}}h=\pi r^2h$. Units are cubic.
  • Lateral area (curved side): $LA=Ch=2\pi rh$.
  • Closed total surface: $SA=2A_{\mathrm{base}}+LA$ (two bases plus curved side).
  • Expanded form: $SA=2\pi r^2+2\pi rh$. Units are square.
A closed cylinder beside its three surface parts: two matching circular ends and one rectangular curved-side piece, shown in matching colors.
Two circular ends plus one curved side make the closed-cylinder surface.

Explain · 4 minSource: Authored

Model: diameter given, radius required

A closed cylinder has diameter $d=10$ cm and height $h=8$ cm. Find exact and approximate volume and total surface area.

  1. The given measure is the diameter, so $r=d/2=10/2=5$ cm.
  2. Volume: $V=\pi r^2h=\pi(5)^2(8)=200\pi\text{ cm}^3\approx628.32\text{ cm}^3$.
  3. Two bases: $2\pi r^2=50\pi\text{ cm}^2$. Curved side: $LA=2\pi rh=80\pi\text{ cm}^2$.
  4. Total: $SA=50\pi+80\pi=130\pi\text{ cm}^2\approx408.41\text{ cm}^2$.
A closed cylinder with one measurement line across the full top diameter and another alongside its full height.
The top line spans the full diameter; the side line marks height.

Model · 5 minSource: Authored

Pacing checkpoint

You try: closed-cylinder volume and area

Student practice

A closed cylinder has radius $r=3$ cm and height $h=10$ cm. Find exact and approximate volume and total surface area. Show the two surface-area contributions.

A single closed cylinder shown in three-quarter view, with its circular base and curved side visible.
Use the radius for the circular base and the height for the cylinder's length.

Practice · 5 minSource: Authored

Check the setup before the arithmetic

Use the radius in both formulas; when the problem gives diameter, divide it by two first.

Volume uses cubic units. Surface area uses square units and counts only the surfaces that are present.

Feedback · 2 minSource: Authored

Pacing checkpoint

Extension: material in a hollow cylinder

A pipe segment has outer radius $R=4$ cm, inner radius $r=3$ cm, and length $h=10$ cm. Find the volume of material.

  1. The material cross-section is an annulus: $A_{\mathrm{material}}=\pi R^2-\pi r^2=\pi(R^2-r^2)$.
  2. Extend that cross-section through length $h$: $V_{\mathrm{material}}=A_{\mathrm{material}}h=\pi(R^2-r^2)h$.
  3. Substitute: $V=\pi(4^2-3^2)(10)=70\pi\text{ cm}^3\approx219.91\text{ cm}^3$.
  4. Subtract the inner circle's area, not the radii; volume uses cubic units.
A hollow cylindrical pipe segment with a clearly visible annular cut end around an open inner bore.
The blue ring is material; the orange opening is the inner volume removed.

Model · 4 minSource: Authored

Exit check: what is being measured?

A closed cylinder has $r=3$ cm and $h=7$ cm. One student labels its capacity $63\pi$ cm$^2$ and its total surface area $60\pi$ cm$^3$. Identify and correct the mix-up.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 2 minSource: Authored