← A clock face is twelve equal 30-degree parts · Angle Facts in a Figure

A clock face is twelve equal 30-degree parts · 12 practice problems

4.MD.C.54.MD.C.64.MD.C.7

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 90 degrees

A clock shows 3:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 3:00; we want the smaller angle between the two hands.

Givens
  • The time is 3:00.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×0=0∘6 \times 0 = 0^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×3+0.5×0=90∘30 \times 3 + 0.5 \times 0 = 90^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣90−0∣=90∘|90 - 0| = 90^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(90, 360−90)=90∘\min(90,\ 360 - 90) = 90^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 90 degrees
4 · Reviewdoes it hold up?

90 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 2 easy answer: 120 degrees

A clock shows 4:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 4:00; we want the smaller angle between the two hands.

Givens
  • The time is 4:00.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×0=0∘6 \times 0 = 0^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×4+0.5×0=120∘30 \times 4 + 0.5 \times 0 = 120^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣120−0∣=120∘|120 - 0| = 120^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(120, 360−120)=120∘\min(120,\ 360 - 120) = 120^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 120 degrees
4 · Reviewdoes it hold up?

120 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 3 easy answer: 2.5 degrees

A clock shows 1:05.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 1:05; we want the smaller angle between the two hands.

Givens
  • The time is 1:05.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×5=30∘6 \times 5 = 30^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×1+0.5×5=32.5∘30 \times 1 + 0.5 \times 5 = 32.5^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣32.5−30∣=2.5∘|32.5 - 30| = 2.5^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(2.5, 360−2.5)=2.5∘\min(2.5,\ 360 - 2.5) = 2.5^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 2.5 degrees
4 · Reviewdoes it hold up?

2.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 4 easy answer: 180 degrees

A clock shows 6:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 6:00; we want the smaller angle between the two hands.

Givens
  • The time is 6:00.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×0=0∘6 \times 0 = 0^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×6+0.5×0=180∘30 \times 6 + 0.5 \times 0 = 180^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣180−0∣=180∘|180 - 0| = 180^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(180, 360−180)=180∘\min(180,\ 360 - 180) = 180^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 180 degrees
4 · Reviewdoes it hold up?

180 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 5 medium answer: 90 degrees

A clock shows 9:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 9:00; we want the smaller angle between the two hands.

Givens
  • The time is 9:00.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×0=0∘6 \times 0 = 0^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×9+0.5×0=270∘30 \times 9 + 0.5 \times 0 = 270^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣270−0∣=270∘|270 - 0| = 270^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(270, 360−270)=90∘\min(270,\ 360 - 270) = 90^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 90 degrees
4 · Reviewdoes it hold up?

90 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 6 medium answer: 175 degrees

A clock shows 8:10.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 8:10; we want the smaller angle between the two hands.

Givens
  • The time is 8:10.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×10=60∘6 \times 10 = 60^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×8+0.5×10=245∘30 \times 8 + 0.5 \times 10 = 245^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣245−60∣=185∘|245 - 60| = 185^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(185, 360−185)=175∘\min(185,\ 360 - 185) = 175^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 175 degrees
4 · Reviewdoes it hold up?

175 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 7 medium answer: 22.5 degrees

A clock shows 2:15.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 2:15; we want the smaller angle between the two hands.

Givens
  • The time is 2:15.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×15=90∘6 \times 15 = 90^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×2+0.5×15=67.5∘30 \times 2 + 0.5 \times 15 = 67.5^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣67.5−90∣=22.5∘|67.5 - 90| = 22.5^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(22.5, 360−22.5)=22.5∘\min(22.5,\ 360 - 22.5) = 22.5^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 22.5 degrees
4 · Reviewdoes it hold up?

22.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 8 medium answer: 100 degrees

A clock shows 7:20.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 7:20; we want the smaller angle between the two hands.

Givens
  • The time is 7:20.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×20=120∘6 \times 20 = 120^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×7+0.5×20=220∘30 \times 7 + 0.5 \times 20 = 220^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣220−120∣=100∘|220 - 120| = 100^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(100, 360−100)=100∘\min(100,\ 360 - 100) = 100^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 100 degrees
4 · Reviewdoes it hold up?

100 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 9 hard answer: 165 degrees

A clock shows 12:30.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 12:30; we want the smaller angle between the two hands.

Givens
  • The time is 12:30.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×30=180∘6 \times 30 = 180^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×0+0.5×30=15∘30 \times 0 + 0.5 \times 30 = 15^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣15−180∣=165∘|15 - 180| = 165^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(165, 360−165)=165∘\min(165,\ 360 - 165) = 165^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 165 degrees
4 · Reviewdoes it hold up?

165 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 10 hard answer: 80 degrees

A clock shows 10:40.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 10:40; we want the smaller angle between the two hands.

Givens
  • The time is 10:40.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×40=240∘6 \times 40 = 240^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×10+0.5×40=320∘30 \times 10 + 0.5 \times 40 = 320^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣320−240∣=80∘|320 - 240| = 80^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(80, 360−80)=80∘\min(80,\ 360 - 80) = 80^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 80 degrees
4 · Reviewdoes it hold up?

80 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 11 hard answer: 97.5 degrees

A clock shows 5:45.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 5:45; we want the smaller angle between the two hands.

Givens
  • The time is 5:45.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×45=270∘6 \times 45 = 270^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×5+0.5×45=172.5∘30 \times 5 + 0.5 \times 45 = 172.5^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣172.5−270∣=97.5∘|172.5 - 270| = 97.5^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(97.5, 360−97.5)=97.5∘\min(97.5,\ 360 - 97.5) = 97.5^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 97.5 degrees
4 · Reviewdoes it hold up?

97.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.
Variant 12 hard answer: 27.5 degrees

A clock shows 11:55.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

12 3 6 9 ?
Show solution
1 · Understandwhat's really being asked

A clock reads 11:55; we want the smaller angle between the two hands.

Givens
  • The time is 11:55.
  • A full clock face is 360 degrees split into 12 equal 30-degree parts.
Unknowns
  • The smaller angle (in degrees) between the hour and minute hands.
Constraints
  • The hour hand also drifts 0.5 degrees for each passing minute.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5
The minute hand sweeps 6 degrees per minute.
6×55=330∘6 \times 55 = 330^\circ
60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6
The hour hand is 30 degrees per hour plus a half-degree drift each minute.
30×11+0.5×55=357.5∘30 \times 11 + 0.5 \times 55 = 357.5^\circ
Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7
The gap between the hands is the difference of the two angles.
∣357.5−330∣=27.5∘|357.5 - 330| = 27.5^\circ
Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7
Keep the smaller of the gap and its 360-degree complement.
min⁡(27.5, 360−27.5)=27.5∘\min(27.5,\ 360 - 27.5) = 27.5^\circ
Two hands cut the circle into two arcs; we want the smaller one.
Answer: 27.5 degrees
4 · Reviewdoes it hold up?

27.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4
  • 4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
  • 4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
  • 4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.
💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.