MATH 130 · Unit 2

Area and Circle Geometry

Area decomposition, circle measures, arcs, sectors, and angle theorems

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    decompose figures and compute area accurately

  2. 2

    use circle terminology and circumference-area formulas

  3. 3

    apply central-angle, inscribed-angle, chord, and arc relationships

Activate · 0.5 minSource: 2

Chapter transition

Session 1 · Area and Circle Geometry

Connect each diagram to the correct radius, arc, chord, or angle relationship.

Activate · 0 minSource: Authored

Predict: decompose figures and compute area accurately

Predict first

Why can the same compound figure be decomposed in more than one valid way?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

decompose figures and compute area accurately

Concept
  • Area is additive for nonoverlapping pieces.
  • Choose rectangles, triangles, and circles that minimize unknown dimensions.

Explain · 2.5 minSource: 3

Model the Skill: decompose figures and compute area accurately

A 15-by-10 rectangle has a 5-by-4 corner removed. Find the remaining area two ways.

  1. Subtraction: $15(10)-5(4)=150-20=130$ square units.
  2. An additive decomposition of the same region also totals 130 square units.
  3. The result is less than the 150-square-unit bounding rectangle.

Model · 3.0 minSource: 4

You Try: decompose figures and compute area accurately

Student practice

Find the area of a 12-by-9 rectangle with a 4-by-3 corner removed.

Practice · 3.0 minSource: Authored

Check and Diagnose: decompose figures and compute area accurately

Error analysis
  • Label units as square units.
  • Check that the result is smaller than the bounding rectangle.
Self-check

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

Feedback · 2.0 minSource: 5

Pacing checkpoint

Make It Stick: decompose figures and compute area accurately

Describe two valid decomposition strategies for a compound region.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: use circle terminology and circumference-area formulas

Predict first

If a radius doubles, by what factor do circumference and area change?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

use circle terminology and circumference-area formulas

Formula focus
  • Circumference: $C=2\pi r$.
  • Area: $A=\pi r^2$.
  • A sector is bounded by two radii and an arc; a segment is bounded by a chord and arc.

Explain · 2.5 minSource: 6

Model the Skill: use circle terminology and circumference-area formulas

1

Circumference changes from $6\pi$ to $12\pi$, so it doubles.

2

Area changes from $9\pi$ to $36\pi$, so it quadruples.

3

The annulus area is $36\pi-9\pi=27\pi$ square units.

Model · 3.0 minSource: 7

You Try: use circle terminology and circumference-area formulas

Student practice

A washer has outer radius 8 and inner radius 5. Find its exact area.

Practice · 3.0 minSource: 8

Check and Diagnose: use circle terminology and circumference-area formulas

Error analysis
  • Use radius, not diameter, in formulas.
  • Subtract circle areas, not radii, for an annulus.
Self-check

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

Feedback · 2.0 minSource: 9

Pacing checkpoint

Make It Stick: use circle terminology and circumference-area formulas

Explain why circle area scales quadratically while circumference scales linearly.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: apply central-angle, inscribed-angle, chord, and arc relationships

Predict first

How does an intercepted arc compare with its central angle and an inscribed angle?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

apply central-angle, inscribed-angle, chord, and arc relationships

1

A central angle has the same measure as its intercepted arc.

2

An inscribed angle measures half its intercepted arc.

3

Equal chords intercept equal arcs.

4

Angles formed by intersecting chords inside a circle use half the sum of the intercepted arcs.

Explain · 2.5 minSource: 10

Model the Skill: apply central-angle, inscribed-angle, chord, and arc relationships

Use a central angle of 72° and its minor arc; then find an inscribed angle intercepting a 110° arc and a chord angle intercepting arcs 80° and 40°.

  1. The central angle and minor arc are both 72°.
  2. The inscribed angle is $110^\circ/2=55^\circ$.
  3. The intersecting-chord angle is $(80^\circ+40^\circ)/2=60^\circ$.
  4. Locate the vertex first; the vertex determines which circle-angle rule applies.

Model · 3.0 minSource: 11

You Try: apply central-angle, inscribed-angle, chord, and arc relationships

Student practice
  1. A central angle and an inscribed angle intercept the same minor arc. If the central angle is 86°, find the arc measure and the inscribed angle.
  2. An inscribed angle measures 37°. Find its intercepted arc.

Practice · 4.0 minSource: Authored

Check and Diagnose: apply central-angle, inscribed-angle, chord, and arc relationships

Identify the vertex location before choosing a theorem.

Do not confuse central and inscribed angles.

Feedback · 2.0 minSource: 12

Pacing checkpoint

Make It Stick: apply central-angle, inscribed-angle, chord, and arc relationships

Use vertex location to decide whether the intercepted arc matches a central angle or is twice an inscribed angle.

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: Authored