2D Mensuration • Topic 3 of 3

Perimeter Problems

Perimeter is the total length of the boundary, and CAT raises the stakes by mixing straight edges with curved ones. The rule is simple: walk the boundary once and add every piece you cross. For a semicircular region the perimeter is the arc πr PLUS the diameter 2r, not just πr — forgetting the straight edge is the most common slip. A "running track" or stadium shape is two straight sides plus two semicircular ends, so its perimeter is 2L + 2πr (the ends together make one full circle). For a sector the perimeter is the arc plus two radii, arc + 2r. When a path of uniform width runs around or inside a region, work with the outer and inner boundaries separately. Equilateral and regular polygons are easy — perimeter is just n × side — but watch the inverse questions where the perimeter is given and you must back out the side or radius before computing an area. Always confirm which edges actually lie on the boundary; an internal diagonal or radius is not part of the perimeter.

✅ Solved examples

1. Find the perimeter of a semicircular region of radius 7 cm (π = 22/7).
Perimeter = πr + 2r = (22/7)(7) + 14 = 22 + 14 = 36 cm. (Arc plus the diameter.)
2. A rectangular field is 40 m by 30 m. Find its perimeter and the cost of fencing it at ₹25 per metre.
Perimeter = 2(40+30) = 140 m. Cost = 140 × ₹25 = ₹3,500.
3. A sector of radius 9 cm subtends 60°. Find its perimeter (in terms of π).
Arc = (60/360)·2π(9) = (1/6)(18π) = 3π. Perimeter = arc + 2r = 3π + 18 ≈ 27.42 cm.
4. A running track has two straight sides of 100 m each and two semicircular ends of radius 35 m. Find the inner perimeter (π = 22/7).
Two semicircles = one full circle = 2π(35) = 2(22/7)(35) = 220 m. Perimeter = 2(100) + 220 = 420 m.

✏️ Practice — try these, take hints as needed

1. Perimeter of a semicircle of radius 14 cm (π = 22/7)?
πr + 2r.
(22/7)(14) + 28.
44 + 28.
72 cm
2. A square has area 144 cm². Find its perimeter.
Side = √144.
Side = 12.
P = 4 × side.
48 cm
3. Perimeter of a sector, radius 6 cm, angle 90° (in terms of π)?
Arc = ¼·2πr = 3π.
Add 2r.
3π + 12.
3π + 12 ≈ 21.42 cm
4. An equilateral triangle has perimeter 36 cm. Find its area.
Side = 36/3 = 12.
(√3/4)a².
a² = 144.
36√3 ≈ 62.35 cm²
5. A wire bent into a circle of radius 7 cm is re-bent into a square. Find the square’s side (π = 22/7).
Wire length = 2πr = 44 cm.
Square perimeter = 44.
Side = 44/4.
11 cm

📝 Topic test — 8 questions

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