MATH 130 · Unit 2

Circle, Radian, and Speed Review

Monday, October 19, 2026 · Unit 2D review of circle diagrams, conversions, arcs, sectors, and rotational speed

55 minutes · Offline capable

Today’s practice sequence

  1. 1

    Read the circle diagram

  2. 2

    Convert the angle and keep precision

  3. 3

    Find the unknown in an arc or sector

  4. 4

    Convert rpm and use the radius

Activate · 0 minSource: Authored

Where is the vertex?

Predict first

Locate O and B. Which angle is central and which is inscribed?

Commit to a response before discussion.

Circle centered at O. Central angle AOD is 124 degrees. Point B lies outside the minor arc AD, with chords BA and BD. Find angle ABD and minor arc AD.
O is the center. Both angles intercept arc AD.

Activate · 0.5 minSource: Authored

Angles and intercepted arcs

Concept
  • A central angle equals the measure of its intercepted arc.
  • An inscribed angle is half its intercepted arc.
  • All arcs around a circle total 360°.
Circle centered at O. Central angle AOD is 124 degrees. Point B lies outside the minor arc AD, with chords BA and BD. Find angle ABD and minor arc AD.
O is the center. Both angles intercept arc AD.

Explain · 1 minSource: Authored

Central and inscribed angles

Find angle ABD and minor arc AD.

  1. The central angle AOD is 124°, so minor arc AD is 124°.
  2. Angle ABD intercepts that same arc: $124^\circ/2=62^\circ$.
  3. The vertex location determines the rule.
Circle centered at O. Central angle AOD is 124 degrees. Point B lies outside the minor arc AD, with chords BA and BD. Find angle ABD and minor arc AD.
O is the center. Both angles intercept arc AD.

Model · 2 minSource: Authored

Triangle and arc practice

Student practice
  1. Find angle BAC and minor arc BC.
  2. Show the intermediate arc or triangle-angle calculation.
Triangle ABC inscribed in a circle. Minor arc AB is 78 degrees and angle ABC is 66 degrees. Find angle BAC and minor arc BC.
Given: arc AB = 78° and angle ABC = 66°.

Practice · 3 minSource: Authored

Two routes to the same angle

Arc route: double the inscribed angle, subtract known arcs from 360°, then halve the remaining arc.

Triangle route: angle ACB is 39°, so angle BAC is 180° − 66° − 39° = 75°.

Feedback · 1 minSource: Authored

Pacing checkpoint

Circle-angle checkpoint

  1. Name the vertex and intercepted arc before calculating.
  2. Central angle = arc. Inscribed angle = half the arc.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Which conversion is needed?

Predict first
  • What changes when an angle is written as DMS, decimal degrees, or radians?
  • Which form is required in $s=r\theta$?

Commit to a response before discussion.

Activate · 0.5 minSource: Authored

Angle conversion rules

Formula focus
  • $1^\circ=60\prime$ and $1\prime=60\prime\prime$.
  • DMS to decimal: $D+M/60+S/3600$.
  • Degrees to radians: multiply by $\pi/180$.
  • Radians to degrees: multiply by $180/\pi$.

Explain · 1 minSource: Authored

Decimal degrees to DMS

Convert 38.726145° to DMS, to the nearest second.

  1. Degrees: 38°. Multiply the decimal part by 60: $0.726145(60)=43.5687$.
  2. Minutes: 43′. Multiply the remaining decimal by 60: $0.5687(60)=34.122$.
  3. Round once, at the end: 38°43′34″.

Model · 2 minSource: Authored

DMS to decimal degrees

Convert 101°18′24.7″ to decimal degrees, to four decimal places.

  1. $101+18/60+24.7/3600=101.3068611\ldots$
  2. Rounded to four decimal places: 101.3069°.

Model · 1.5 minSource: Authored

Decimal degrees to radians

Convert 137.6° to radians, to three decimal places.

  1. $137.6(\pi/180)=2.401572\ldots$
  2. Keep π in the calculator expression. Round the final answer: 2.402 rad.

Model · 1 minSource: Authored

Conversion practice

Student practice
  1. Convert 58.417862° to DMS, to the nearest second.
  2. Convert 112°36′18.5″ to decimal degrees, to four decimal places.

Practice · 3 minSource: Authored

Precision and notation

Error analysis
  • Minutes and seconds use base 60. A decimal degree is not DMS notation.
  • Match the question’s requested decimal places. Round the final result once.
Self-check

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

Feedback · 0.5 minSource: Authored

Pacing checkpoint

Conversion checkpoint

  1. DMS to decimal uses division by 60 and 3600.
  2. Decimal to DMS uses the fractional part twice, multiplying by 60.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Length or area?

Predict first
  • The same radius and angle determine both a curved distance and a shaded area.
  • Which quantity uses square units?

Commit to a response before discussion.

A circular sector with radius 6 units and central angle 1.4 radians. The curved boundary is marked s and the region A.
The curved boundary is a length; the shaded region is an area.

Activate · 1 minSource: Authored

Arc and sector formulas

Formula focus
  • Arc length: $s=r\theta$. Sector area: $A=\tfrac12r^2\theta$.
  • Both formulas require θ in radians.
  • Unknown radius: $r=s/\theta$.
  • Unknown sector angle: $\theta=2A/r^2$, then convert to degrees if requested.

Explain · 2 minSource: Authored

One sector, two measurements

Radius = 6 units and angle = 1.4 rad. Find arc length and sector area.

  1. $s=6(1.4)=8.4$ units.
  2. $A=\tfrac12(6^2)(1.4)=25.2$ square units.
  3. The radius is squared only in the area formula.
A circular sector with radius 6 units and central angle 1.4 radians. The curved boundary is marked s and the region A.
The curved boundary is a length; the shaded region is an area.

Model · 2 minSource: Authored

Solve backwards for radius

Student practice
  1. Together: an arc is 24 units long with central angle 80°. Find the radius.
  2. Then independently: an arc is 21 units long with central angle 105°. Find the radius.

Practice · 3 minSource: Authored

Solve backwards for angle

A sector has area 35.4 square units and radius 4 units. Find θ in radians and degrees.

  1. $\theta=2A/r^2=2(35.4)/4^2=4.425$ rad.
  2. $\theta_{deg}=4.425(180/\pi)\approx253.53^\circ$.
  3. The angle is between π and 2π radians, so a sector larger than a semicircle is reasonable.
A sector larger than a semicircle has radius 4 units and area 35.4 square units. Its central angle is unknown.
The highlighted sector can have an angle above 180°.

Model · 2 minSource: Authored

Check an inverse-sector setup

Student practice
  1. Without looking at the model, write an equation for a sector with area 35.4 and radius 4.
  2. Solve for the angle in radians and degrees.
  3. Explain why an angle above 180° is allowed.
A sector larger than a semicircle has radius 4 units and area 35.4 square units. Its central angle is unknown.
The highlighted sector can have an angle above 180°.

Practice · 3 minSource: Authored

Check the setup before arithmetic

  • For radius, divide arc length by the angle in radians.
  • For angle, double the sector area and divide by the squared radius.

A 150° sector with radius 20 has area $500\pi/3\approx523.60$ square units.

Feedback · 3 minSource: Authored

Pacing checkpoint

Arc and sector checkpoint

  1. Circle the requested unknown first.
  2. Convert degrees to radians before substitution.
  3. Check whether the answer is a length, an area, or an angle.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1 minSource: Authored

Same rotation, different distance

Predict first
  • Do the center and the rim have the same angular speed?
  • Does a point farther from the center travel farther per turn?

Commit to a response before discussion.

Activate · 0.5 minSource: Authored

From rpm to speed

Formula focus
  • Use radius = diameter ÷ 2.
  • Degrees per second: $n\cdot360/60$.
  • Radians per second: $\omega=n\cdot2\pi/60$.
  • Linear speed: $v=r\omega$ in distance units per second.

Explain · 1.5 minSource: Authored

A rotating wheel

Diameter = 12 ft, rotation = 24 rpm. Find angular speed in degrees/s and rim speed in ft/s.

  1. $r=12/2=6$ ft.
  2. $24(360)/60=144^\circ/\mathrm{s}$.
  3. $\omega=24(2\pi)/60=4\pi/5$ rad/s.
  4. $v=6(4\pi/5)=24\pi/5\approx15.08$ ft/s.
A wheel with diameter 12 feet makes 24 revolutions per minute. Find angular speed in degrees per second and rim speed in feet per second.
The given length crosses the entire wheel.

Model · 3 minSource: Authored

Rotation practice

Student practice
  1. A wheel has diameter 8 ft and turns at 30 rpm.
  2. Find angular speed in degrees/s and linear speed at the rim in ft/s.
  3. Write the diameter-to-radius step and both unit conversions.

Practice · 4 minSource: Authored

Two common speed errors

Error analysis
  • Using the diameter in $v=r\omega$ doubles the speed.
  • Degrees/s cannot go directly into $v=r\omega$; use radians/s.
  • Convert minutes to seconds once, and label the final units.
Self-check

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

Feedback · 2 minSource: Authored

Pacing checkpoint

Speed checkpoint

  1. Radius first. Convert rpm. Then multiply radius by radians per second.
  2. A sensible answer includes a distance unit and a time unit.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1 minSource: Authored

Independent exit check

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

Synthesize · 3 minSource: Authored