Diameter is twice the radius

3.OA.C.73.G.A.1 · adapt · grade 3

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

In the figure on the right, the diameter of the largest circle is $40\ \text{cm}$ . What is the radius of the smallest circle, in cm?

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Understand

Three circles of different sizes all touch at one point on the left and share the same horizontal diameter line. The medium circle is nested inside the largest, and the smallest is nested inside the medium. The largest circle's diameter is 40 cm. We must find the radius of the smallest circle.

Givens

The largest circle has diameter 40 cm.
All three circles touch at the same single point on the left end.
Their centers all lie on one horizontal line; the medium circle fits inside the largest, and the smallest fits inside the medium.
In the figure the radii of the three circles are in the ratio 100 : 60 : 30 (largest : medium : smallest).

Unknowns

The radius of the smallest circle in centimeters.

Constraints

Diameter is always twice the radius (diameter = radius x 2).
Because the circles are tangent at the same left point and nested, their sizes follow the drawn proportions: medium radius is 3/5 of the largest, smallest radius is half of the medium.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Read the nested-circle picture to see the relative sizes, then break the work into small steps: first the largest radius from its diameter, then the medium radius, then the smallest radius, using diameter = 2 x radius each time.

Execute

#7 Identify Subproblems 3.G.A.1

The largest circle's diameter is 40 cm. Radius is half the diameter.

40 \div 2 = 20

Diameter is twice the radius, so halving the diameter gives the radius.

#1 Draw a Diagram 3.OA.C.7

From the figure the medium circle's radius is 3/5 of the largest circle's radius (drawn as 60 next to 100). So the medium radius is 20 split into fifths, taken three times.

20 \div 5 \times 3 = 4 \times 3 = 12

Reading the matching proportions in the picture lets a Grade 3 student scale the known radius down.

#1 Draw a Diagram 3.OA.C.7

From the figure the smallest circle's radius is half the medium circle's radius (drawn as 30 next to 60). So halve the medium radius.

12 \div 2 = 6

Half of the medium radius gives the smallest radius, a simple division within 100.

Answer: 6 cm

Review

The smallest radius (6 cm) is far less than the largest radius (20 cm), which matches a tiny circle nested inside two bigger ones. Units are centimeters, correct for a radius.

Track diameters instead: largest 40 cm, medium 24 cm, smallest 12 cm, then halve the smallest diameter: 12 / 2 = 6 cm (Tool 8, Analyze the Units).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using the diameter = twice the radius relationship for the largest circle.
3.OA.C.7 Fluently multiply and divide within 100 — Scaling the radii (20 -> 12 -> 6) by multiplying and dividing within 100.

💡 Half a diameter is a radius -- keep halving and scaling with Grade 3 division and the smallest circle pops out!