Perimeter of shapes from tangent circles

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 circles of two different sizes drawn so that neighboring circles touch. The large circles each have a diameter of $8\text{ in}$ and the small circles each have a diameter of $5\text{ in}$ . The center of each circle is joined to the centers of its neighbors to form a quadrilateral. What is the perimeter of the quadrilateral, in inches? (The two large circles are the same size as each other, and the two small circles are the same size as each other.)

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Understand

Two large circles (diameter 8 in each) sit on the left and right, and two small circles (diameter 5 in each) sit on top and bottom, all touching around a central gap. Joining the centers of neighboring circles makes a four-sided figure (a rhombus). Each side connects a large circle's center to a small circle's center. We must find the perimeter of the quadrilateral.

Givens

Two large circles, each with diameter 8 in (so radius 4 in).
Two small circles, each with diameter 5 in (so radius 2.5 in).
Neighboring circles touch (are tangent).
Each of the four sides joins the center of a large circle to the center of a small circle.

Unknowns

The perimeter of the quadrilateral in inches.

Constraints

When a large and a small circle touch on the outside, the distance between their centers is the large radius plus the small radius.
All four sides join a large center to a small center, so all four sides are equal.

Plan

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

Find one side as large radius + small radius using the touching rule, then multiply by the four equal sides to get the perimeter.

Execute

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

Radius is half the diameter. Large radius is 8 / 2 = 4 in; small radius is 5 / 2 = 2.5 in.

8 \div 2 = 4,\quad 5 \div 2 = 2.5

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

#7 Identify Subproblems 3.MD.D.8

Where a large and small circle touch, the touch point is one large radius from the large center and one small radius from the small center. So the center-to-center distance is large radius + small radius.

4 + 2.5 = 6.5

Two radii laid end to end across the touch point give the distance between the two centers.

#7 Identify Subproblems 3.OA.C.7

All four sides each join a large center to a small center, so each side is 6.5 in. The perimeter is four of them.

4 \times 6.5 = 26

Four equal sides are totaled by multiplication, a Grade 3 idea, with a simple half-inch piece.

Answer: 26 in

Review

Each side 6.5 in is between the large radius (4) and the full diameter (8), which is sensible for a center-to-center distance. Four sides give 26 in. Units are inches, correct for a perimeter.

Add the sides one at a time: 6.5 + 6.5 + 6.5 + 6.5 = 26 in (Tool 2, Make a Systematic List).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using diameter = twice radius to get the large (4 in) and small (2.5 in) radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding one side as large radius + small radius and totaling the perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 4 x 6.5 = 26.

💡 Each side is a big radius plus a small radius -- add them, then multiply by 4 sides with Grade 3 math!