Perimeter of shape joining circle centers

3.MD.D.83.OA.C.7 · adapt · grade 3

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

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $36\text{ in}$ , what is the diameter of one circle, in inches?

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Understand

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 36 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 36 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

Plan

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

Execute

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

#7 Identify Subproblems 3.OA.C.7

Three diameters make 36 inches, so divide by 3.

36 \div 3 = 12

Sharing 36 equally among 3 equal sides is division within 100.

Answer: 12 in

Review

A diameter of 12 inches gives a triangle perimeter of 3 x 12 = 36 inches, matching the given. Units are inches, correct for a length.

Use 3 x d = 36 and solve for the unknown d: d = 36 / 3 = 12 in (Tool 13, Convert to Algebra).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 36 / 3 = 12.

💡 Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!