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
convert angle measures and compute arc length and sector area
2
relate angular velocity, linear velocity, and distance
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.
For inverse problems, solve $r=s/\theta$ or $\theta=2A/r^2$ before substituting.
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

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

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

$A=\tfrac12r^2\theta=48\pi\approx150.796$ 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°.
An arc has length $6\pi$ units and central angle $2\pi/5$ radians. Solve for the radius.
A sector has radius 5 units and area $25\pi/6$ square units. Find its central angle in radians and degrees.
Convert 135° to radians.
Convert $67.425^\circ$ to DMS and $22^\circ18'36''$ to decimal degrees.

Feedback and solution

$40^\circ=2\pi/9$ radians, so $s=9(2\pi/9)=2\pi$.
$r=s/\theta=6\pi/(2\pi/5)=15$ units.
$25\pi/6=\tfrac12(5^2)\theta$ gives $\theta=\pi/3=60^\circ$.
$135^\circ=3\pi/4$ radians.
$67.425^\circ=67^\circ25'30''$; $22^\circ18'36''=22.31^\circ$.

Practice · 6.0 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.
For an arc, isolate $r=s/\theta$ when radius is unknown; for a sector, isolate $\theta=2A/r^2$.
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 how you would recover a radius from arc length and angle, then recover a central angle from sector area and radius.

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: $d=vt=r\theta$.
For rpm, multiply by 360°/60 to get degrees per second; use 2π/60 to get radians per second for $v=rω$.

Explain · 2.5 minSource: 23

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

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

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

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.
A wheel of radius 3 ft turns at 18 revolutions per minute. Find angular speed in degrees per second and rim speed in feet per second.

Feedback and solution

$v=r\omega=0.35(8)=2.8$ m/s.
$18(360)/60=108^\circ/s$; $108^\circ/s=3\pi/5$ rad/s, so $v=3(3\pi/5)=9\pi/5\approx5.7$ ft/s.

Practice · 5.0 minSource: Authored

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

Error analysis

Angular units must be radians for $v=r\omega$.
For rev/min, convert to degrees per second with 360/60 and radians per second with 2π/60 before using $v=rω$.
Check that linear speed units include distance per time.

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 to convert rpm to degrees per second and radians per second, then use radius to find rim 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

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

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

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

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

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.

Feedback and solution

$V=90\pi$ cubic units.
$SA=78\pi$ square units.

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

Circle review: identify central versus inscribed angles and determine the intercepted arc.
For Test 2 practice, compute arcs and sectors, convert angle measures, and relate angular and linear speed.
Record one Unit 2D topic to revisit before Test 2.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 29, 30