Polygon side equals sum of two radii

3.MD.D.83.G.A.1 · adapt · grade 3

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

The figure on the right shows four circles of different sizes drawn so that they touch one another in a ring, with the centers of the circles joined to form quadrilateral ABCD. If the perimeter of quadrilateral ABCD is $60\ \text{cm}$ , what is the sum of the radii of the four circles, in centimeters?

Show solution

Understand

Four circles of different sizes touch one another in a ring. Joining their centers makes a quadrilateral ABCD. Each side connects two touching circles, so each side length equals the sum of those two circles' radii. The quadrilateral's perimeter is 60 cm, and I need the total of all four radii.

Givens

Four circles touch (are tangent) in a ring; neighboring circles touch.
Centers joined form quadrilateral ABCD.
Each side equals the sum of the radii of the two circles meeting along it.
Perimeter of ABCD is 60 cm.

Unknowns

The sum of the radii of the four circles, in cm.

Constraints

When two circles touch on the outside, the distance between their centers equals the sum of their radii.
Every radius is counted in exactly two sides (each circle touches two neighbors).

Plan

#9 Solve an Easier Related Problem · also uses: #1 Draw a Diagram

Write each side as a radius sum, then add all four sides. Looking at the sum reveals each radius appears exactly twice, turning the perimeter into double the radius total — a simpler relationship than handling each unknown radius separately.

Execute

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

Two touching circles have their centers separated by the sum of their radii. So AB = rA + rB, BC = rB + rC, CD = rC + rD, and DA = rD + rA.

AB+BC+CD+DA = (r_A+r_B)+(r_B+r_C)+(r_C+r_D)+(r_D+r_A)

The figure shows each side bridging two circles that touch, so the side is just both radii laid end to end.

#9 Solve an Easier Related Problem 3.OA.D.9

When the four sides are added, every circle's radius appears exactly two times, because each circle touches two neighbors. So the perimeter equals twice the total of all four radii.

60 = 2 \times (r_A+r_B+r_C+r_D)

Spotting the repeated pattern of '+each radius twice' makes the big sum collapse into a simple doubling.

#9 Solve an Easier Related Problem 3.OA.A.2

Since the perimeter is double the radius total, divide 60 by 2 to get the sum of the radii.

60 \div 2 = 30

Undoing a doubling is just dividing by 2, a basic grade-3 division fact.

Answer: 30 cm

Review

The radius total (30 cm) is exactly half the 60-cm perimeter, which makes sense because the perimeter counts every radius twice. Units stay in centimeters, and 30 is a reasonable size compared with the 60-cm boundary.

Guess and check (tool 6): pick any four radii that sum to 30, such as 6, 8, 7, 9; the four sides 14, 15, 16, 15 add to 60, confirming the rule.

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing each side of the center-quadrilateral as a sum of two touching circles' radii.
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that each radius appears twice when all sides are added, so the perimeter is double the radius total.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Dividing the perimeter by 2 to recover the sum of the radii.

💡 This only needs Grade 3 thinking: every radius gets counted twice, so just split the perimeter in half!