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Perimeter of a shape made from three semicircular arcs

Add arcs, not the dashed diameter

A semicircular arc with diameter $d$ has length $π d /2$ . If the two small diameters add to the large diameter $D$ , the three arcs together have length $π D$ . A dashed guide line is not part of the perimeter unless the problem explicitly says it is.

Problem

A closed outline consists of a large semicircular arc above a dashed line and two equal semicircular arcs below it. Each small semicircle has diameter $9$ m; together their diameters span the large semicircle. Find the perimeter exactly and to two decimal places.

Answer

18 π

m, or approximately

56.55

The dashed diameter is only a construction guide and is not counted as a boundary.

Step-by-step solution

1. Find the large diameter

The two small diameters are $9$ m each, so the large diameter is $D = 9 + 9 = 18$ m.

2. Add the three arc lengths

The large arc has length $18 π /2 = 9 π$ m. Each small arc has length $9 π /2 = 4.5 π$ m. Hence

P = 9 π + 4.5 π + 4.5 π = 18 π m .

3. Round if requested

18 π \approx 56.55 m .

The same result follows directly from the rule $P = π D$ for this particular arrangement.

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