Find each radius and diameter from segments

3.OA.C.73.G.A.1 · adapt · grade 3

Archetype: Radius and Diameter Relationships · step in a 11-type progression

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_{\text{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_{\text{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_{\text{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 10, 12, 14 in are easy to compare; 10 in is the smallest, so Noah's circle is smallest. The trap was Mia's 7, which is only a radius (14 in across), not the smallest.

Convert everything to radius instead: Liam 6 in, Mia 7 in, Noah 5 in; the smallest radius (5 in, Noah) 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!