← 3-2 · Find each radius and diameter from segments · Radius and Diameter Relationships

Find each radius and diameter from segments · 12 practice problems

3.G.A.13.OA.C.73.OA.A.3

Generated variants — 12

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

Variant 1 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $22\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $6\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $20\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 22 in. Mia set her compass so the point and pencil were 6 in apart. Noah's segment that splits his circle into two equal halves is 20 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 22 in.
Mia: the compass opening (point to pencil) is 6 in.
Noah: the segment that splits the circle into two equal halves is 20 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 22 in.

d_{Liam} = 22

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 6 \times 2 = 12

Diameter is twice the radius, so double the 6 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 20 in.

d_{Noah} = 20

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 22 in, Mia 12 in, Noah 20 in. The smallest diameter belongs to the smallest circle.

12 < 20 < 22

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 22, 12, 20 in are easy to compare; 12 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (12 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 11 in, Mia 6 in, Noah 10 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 6 in radius to a 12 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 2 answer: Noah

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $18\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $9\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $16\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 18 in. Mia set her compass so the point and pencil were 9 in apart. Noah's segment that splits his circle into two equal halves is 16 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 18 in.
Mia: the compass opening (point to pencil) is 9 in.
Noah: the segment that splits the circle into two equal halves is 16 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 18 in.

d_{Liam} = 18

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 9 \times 2 = 18

Diameter is twice the radius, so double the 9 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 16 in.

d_{Noah} = 16

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 18 in, Mia 18 in, Noah 16 in. The smallest diameter belongs to the smallest circle.

16 < 18 < 18

With the same measure for all three, the smallest number is the smallest circle.

Answer: Noah

Review

After making every clue a diameter, the values 18, 18, 16 in are easy to compare; 16 in is the smallest, so Noah's circle is smallest. Mia's clue is only a radius (18 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 9 in, Mia 9 in, Noah 8 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 9 in radius to a 18 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 3 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $14\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $5\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $12\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 14 in. Mia set her compass so the point and pencil were 5 in apart. Noah's segment that splits his circle into two equal halves is 12 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 14 in.
Mia: the compass opening (point to pencil) is 5 in.
Noah: the segment that splits the circle into two equal halves is 12 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 14 in.

d_{Liam} = 14

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 5 \times 2 = 10

Diameter is twice the radius, so double the 5 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 12 in.

d_{Noah} = 12

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 14 in, Mia 10 in, Noah 12 in. The smallest diameter belongs to the smallest circle.

10 < 12 < 14

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 14, 10, 12 in are easy to compare; 10 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (10 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 7 in, Mia 5 in, Noah 6 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 5 in radius to a 10 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 4 answer: Noah

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $16\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $9\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $12\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 16 in. Mia set her compass so the point and pencil were 9 in apart. Noah's segment that splits his circle into two equal halves is 12 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 16 in.
Mia: the compass opening (point to pencil) is 9 in.
Noah: the segment that splits the circle into two equal halves is 12 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 16 in.

d_{Liam} = 16

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 9 \times 2 = 18

Diameter is twice the radius, so double the 9 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 12 in.

d_{Noah} = 12

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 16 in, Mia 18 in, Noah 12 in. The smallest diameter belongs to the smallest circle.

12 < 16 < 18

With the same measure for all three, the smallest number is the smallest circle.

Answer: Noah

Review

After making every clue a diameter, the values 16, 18, 12 in are easy to compare; 12 in is the smallest, so Noah's circle is smallest. Mia's clue is only a radius (18 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 8 in, Mia 9 in, Noah 6 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 9 in radius to a 18 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 5 answer: Liam

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $10\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $8\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $14\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 10 in. Mia set her compass so the point and pencil were 8 in apart. Noah's segment that splits his circle into two equal halves is 14 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 10 in.
Mia: the compass opening (point to pencil) is 8 in.
Noah: the segment that splits the circle into two equal halves is 14 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 10 in.

d_{Liam} = 10

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 8 \times 2 = 16

Diameter is twice the radius, so double the 8 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 14 in.

d_{Noah} = 14

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 10 in, Mia 16 in, Noah 14 in. The smallest diameter belongs to the smallest circle.

10 < 14 < 16

With the same measure for all three, the smallest number is the smallest circle.

Answer: Liam

Review

After making every clue a diameter, the values 10, 16, 14 in are easy to compare; 10 in is the smallest, so Liam's circle is smallest. Mia's clue is only a radius (16 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 5 in, Mia 8 in, Noah 7 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 8 in radius to a 16 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 6 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $20\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $8\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $18\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 20 in. Mia set her compass so the point and pencil were 8 in apart. Noah's segment that splits his circle into two equal halves is 18 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 20 in.
Mia: the compass opening (point to pencil) is 8 in.
Noah: the segment that splits the circle into two equal halves is 18 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 20 in.

d_{Liam} = 20

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 8 \times 2 = 16

Diameter is twice the radius, so double the 8 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 18 in.

d_{Noah} = 18

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 20 in, Mia 16 in, Noah 18 in. The smallest diameter belongs to the smallest circle.

16 < 18 < 20

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 20, 16, 18 in are easy to compare; 16 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (16 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 10 in, Mia 8 in, Noah 9 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 8 in radius to a 16 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 7 answer: Noah

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $12\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $7\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $10\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 12 in. Mia set her compass so the point and pencil were 7 in apart. Noah's segment that splits his circle into two equal halves is 10 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 12 in.
Mia: the compass opening (point to pencil) is 7 in.
Noah: the segment that splits the circle into two equal halves is 10 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 12 in.

d_{Liam} = 12

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 7 \times 2 = 14

Diameter is twice the radius, so double the 7 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 10 in.

d_{Noah} = 10

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 12 in, Mia 14 in, Noah 10 in. The smallest diameter belongs to the smallest circle.

10 < 12 < 14

With the same measure for all three, the smallest number is the smallest circle.

Answer: Noah

Review

After making every clue a diameter, the values 12, 14, 10 in are easy to compare; 10 in is the smallest, so Noah's circle is smallest. Mia's clue is only a radius (14 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 6 in, Mia 7 in, Noah 5 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 7 in radius to a 14 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 8 answer: Liam

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $12\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $9\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $16\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 12 in. Mia set her compass so the point and pencil were 9 in apart. Noah's segment that splits his circle into two equal halves is 16 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 12 in.
Mia: the compass opening (point to pencil) is 9 in.
Noah: the segment that splits the circle into two equal halves is 16 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 12 in.

d_{Liam} = 12

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 9 \times 2 = 18

Diameter is twice the radius, so double the 9 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 16 in.

d_{Noah} = 16

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 12 in, Mia 18 in, Noah 16 in. The smallest diameter belongs to the smallest circle.

12 < 16 < 18

With the same measure for all three, the smallest number is the smallest circle.

Answer: Liam

Review

After making every clue a diameter, the values 12, 18, 16 in are easy to compare; 12 in is the smallest, so Liam's circle is smallest. Mia's clue is only a radius (18 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 6 in, Mia 9 in, Noah 8 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 9 in radius to a 18 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 9 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $24\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $5\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $22\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 24 in. Mia set her compass so the point and pencil were 5 in apart. Noah's segment that splits his circle into two equal halves is 22 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 24 in.
Mia: the compass opening (point to pencil) is 5 in.
Noah: the segment that splits the circle into two equal halves is 22 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 24 in.

d_{Liam} = 24

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 5 \times 2 = 10

Diameter is twice the radius, so double the 5 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 22 in.

d_{Noah} = 22

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 24 in, Mia 10 in, Noah 22 in. The smallest diameter belongs to the smallest circle.

10 < 22 < 24

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 24, 10, 22 in are easy to compare; 10 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (10 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 12 in, Mia 5 in, Noah 11 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 5 in radius to a 10 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 10 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $18\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $7\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $16\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 18 in. Mia set her compass so the point and pencil were 7 in apart. Noah's segment that splits his circle into two equal halves is 16 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 18 in.
Mia: the compass opening (point to pencil) is 7 in.
Noah: the segment that splits the circle into two equal halves is 16 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 18 in.

d_{Liam} = 18

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 7 \times 2 = 14

Diameter is twice the radius, so double the 7 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 16 in.

d_{Noah} = 16

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 18 in, Mia 14 in, Noah 16 in. The smallest diameter belongs to the smallest circle.

14 < 16 < 18

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 18, 14, 16 in are easy to compare; 14 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (14 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 9 in, Mia 7 in, Noah 8 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 7 in radius to a 14 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 11 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $20\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $6\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $14\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 20 in. Mia set her compass so the point and pencil were 6 in apart. Noah's segment that splits his circle into two equal halves is 14 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 20 in.
Mia: the compass opening (point to pencil) is 6 in.
Noah: the segment that splits the circle into two equal halves is 14 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 20 in.

d_{Liam} = 20

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 6 \times 2 = 12

Diameter is twice the radius, so double the 6 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 14 in.

d_{Noah} = 14

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 20 in, Mia 12 in, Noah 14 in. The smallest diameter belongs to the smallest circle.

12 < 14 < 20

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 20, 12, 14 in are easy to compare; 12 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (12 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 10 in, Mia 6 in, Noah 7 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 6 in radius to a 12 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!

Variant 12 answer: Mia

Three children each drew a circle and then described the circle they drew. Which child drew the smallest circle?

Liam: I drew the longest line segment that fits inside my circle, and it was $16\text{ in}$ long.
Mia: I set my compass so the point and the pencil tip were $6\text{ in}$ apart, then drew my circle.
Noah: The line segment that splits my circle into two equal halves is $14\text{ in}$ long.

Show solution

Understand

Three children each drew a circle and described it differently. Liam's longest segment inside his circle is 16 in. Mia set her compass so the point and pencil were 6 in apart. Noah's segment that splits his circle into two equal halves is 14 in. We must decide whose circle is smallest.

Givens

Liam: the longest segment that fits inside a circle is 16 in.
Mia: the compass opening (point to pencil) is 6 in.
Noah: the segment that splits the circle into two equal halves is 14 in.

Unknowns

Which child drew the smallest circle.

Constraints

The longest segment inside a circle is its diameter.
The compass opening is the radius.
A segment splitting a circle into two equal halves passes through the center, so it is the diameter.

Plan

#15 Organize Information in More Ways · also uses: #3 Eliminate Possibilities

Each clue describes either a radius or a diameter. Convert all three to the same measure (diameter), then compare to find the smallest, ruling out the larger circles.

Execute

#15 Organize Information in More Ways 3.G.A.1

The longest segment that fits inside a circle goes through the center, so it is the diameter. Liam's diameter is 16 in.

d_{Liam} = 16

The widest line across a circle is the diameter.

#15 Organize Information in More Ways 3.OA.C.7

The compass opening from point to pencil is the radius (center to edge). Double it to get the diameter.

d_{Mia} = 6 \times 2 = 12

Diameter is twice the radius, so double the 6 in compass opening.

#15 Organize Information in More Ways 3.G.A.1

A segment that splits a circle into two equal halves passes through the center, so it is the diameter. Noah's diameter is 14 in.

d_{Noah} = 14

Only a line through the center divides a circle into two equal halves; that line is the diameter.

#3 Eliminate Possibilities 3.OA.A.3

Now all three are diameters: Liam 16 in, Mia 12 in, Noah 14 in. The smallest diameter belongs to the smallest circle.

12 < 14 < 16

With the same measure for all three, the smallest number is the smallest circle.

Answer: Mia

Review

After making every clue a diameter, the values 16, 12, 14 in are easy to compare; 12 in is the smallest, so Mia's circle is smallest. Mia's clue is only a radius (12 in across), so it must be doubled before comparing.

Convert everything to radius instead: Liam 8 in, Mia 6 in, Noah 7 in; the smallest radius again gives the smallest circle (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Identifying the longest inside segment and the halving segment as diameters.
3.OA.C.7 Fluently multiply and divide within 100 — Doubling Mia's 6 in radius to a 12 in diameter.
3.OA.A.3 Solve multiplication and division word problems within 100 — Comparing the three diameters to choose the smallest circle.

💡 Turn every clue into a diameter first -- once they match, the smallest number wins, just Grade 3 comparing!