MATH 130 · Unit 2

Radians, Velocity, and Solid Geometry

Angle conversion, arc length, sectors, motion, volume, and surface area

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    convert angle measures and compute arc length and sector area

  2. 2

    relate angular velocity, linear velocity, and distance

  3. 3

    select volume and surface-area formulas for solids

Activate · 0.5 minSource: Authored

Chapter transition

Session 2 · Radians, Motion, and Solids

Track units carefully as angles produce distances, speeds, areas, and volumes.

Activate · 0 minSource: Authored

Predict: convert angle measures and compute arc length and sector area

Predict first

Why must the angle be in radians when using $s=r\theta$?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

convert angle measures and compute arc length and sector area

Formula focus
  • Radians measure arc length per radius.
  • $s=r\theta$ and $A=\tfrac12r^2\theta$ require radians.
  • Convert degrees using $\theta_{rad}=\theta_{deg}\pi/180$.

Explain · 2.5 minSource: 14, 15

Model the Skill: convert angle measures and compute arc length and sector area

1

$120^\circ=2\pi/3$ radians.

2

$s=r\theta=12(2\pi/3)=8\pi\approx25.133$ units.

3

$A=\tfrac12r^2\theta=48\pi\approx150.796$ square units.

Model · 3.0 minSource: 16, 17

You Try: convert angle measures and compute arc length and sector area

Student practice

Find the arc length for radius 9 and central angle 40°.

Practice · 3.0 minSource: 18

Check and Diagnose: convert angle measures and compute arc length and sector area

Error analysis
  • Do not insert degrees directly into radian formulas.
  • Sector area should be the same fraction of $\pi r^2$ as the angle is of a full rotation.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 19, 20

Pacing checkpoint

Make It Stick: convert angle measures and compute arc length and sector area

Explain the unit check that confirms an arc-length calculation.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 21

Predict: relate angular velocity, linear velocity, and distance

Predict first

Why do points farther from the center move faster even when angular speed is the same?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

relate angular velocity, linear velocity, and distance

Formula focus
  • Angular speed: $\omega=\theta/t$.
  • Linear speed: $v=r\omega$.
  • Distance: $d=vt=r\theta$.

Explain · 2.5 minSource: 23

Model the Skill: relate angular velocity, linear velocity, and distance

1

$v=r\omega=4(3)=12$ ft/s.

2

$d=vt=12(10)=120$ ft.

3

Radians are dimensionless in the linear-speed conversion.

Model · 3.0 minSource: 24

You Try: relate angular velocity, linear velocity, and distance

Student practice

A wheel of radius 0.35 m turns at 8 rad/s. Find rim speed.

Practice · 3.0 minSource: Authored

Check and Diagnose: relate angular velocity, linear velocity, and distance

Error analysis
  • Angular units must be radians for $v=r\omega$.
  • Distinguish revolutions per minute from radians per second.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 25

Pacing checkpoint

Make It Stick: relate angular velocity, linear velocity, and distance

Describe how radius affects linear speed at fixed angular speed.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: select volume and surface-area formulas for solids

Predict first

Which dimensions contribute to volume, and which surfaces contribute to surface area?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

select volume and surface-area formulas for solids

1

Volume measures three-dimensional capacity; surface area measures exposed two-dimensional faces.

2

Cylinder: $V=\pi r^2h$ and $SA=2\pi r^2+2\pi rh$.

Explain · 2.5 minSource: 27

Model the Skill: select volume and surface-area formulas for solids

1

$V=\pi(R^2-r^2)h$.

2

$V=\pi(4^2-3^2)(10)=70\pi$ cubic units.

3

Subtract the inner cylinder from the outer cylinder.

Model · 3.0 minSource: Authored

You Try: select volume and surface-area formulas for solids

Student practice

Find the volume of a cylinder with radius 3 and height 10, then its total surface area.

Practice · 3.0 minSource: 28

Check and Diagnose: select volume and surface-area formulas for solids

Use cubic units for volume and square units for area.

A pipe requires outer volume minus inner volume.

Feedback · 2.0 minSource: Authored

Pacing checkpoint

Make It Stick: select volume and surface-area formulas for solids

Explain how a diagram determines whether to add, subtract, or omit circular bases.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 29, 30