Rectangle sides are multiples of the diameter

3.OA.C.73.G.A.1 · adapt · grade 3

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

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the square on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

Understand

Inside a square that is 6 cm on each side, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The square has side length 6 cm.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the square.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the 6 cm by 6 cm square and may touch but not overlap.

Plan

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a 2 cm square cell. Drawing the square as a grid of 2 cm cells shows how many cells fit across and down, and the rows-times-columns pattern gives the total.

Execute

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

#5 Look for a Pattern 3.OA.C.7

Along one side, 6 cm holds 6 / 2 = 3 cells. The square is 6 cm tall too, so it holds 3 rows of cells.

6 \div 2 = 3

Splitting a length into equal 2 cm pieces is division within 100.

#10 Create a Physical Representation 3.OA.C.7

A 3 by 3 grid of cells holds one circle per cell.

3 \times 3 = 9

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 9

Review

Nine circles is a whole number, as it must be for counting objects. A 6 cm square has area 36 square cm and each 2 cm cell is 4 square cm, giving 36 / 4 = 9 cells, which agrees.

Lay out the circles in a simple square grid physically: place them in a 3-by-3 arrangement and count 9 directly (Tool 2, Make a Systematic List).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 6 / 2 = 3 per side and 3 x 3 = 9 total.

💡 Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!