The perimeter of a composite shape with curved edges is the total distance around its boundary, including both straight sides and arcs. Work out each arc length using the formula arc length = (angle/360) × πd, then add all straight edges. Keep π in your answer until the final step.
What is the arc length formula?
An arc is a portion of a circle's circumference. The length of an arc depends on two things: the radius of the circle and the angle at the centre that the arc subtends.
Arc length = (θ/360) × πd = (θ/360) × 2πr
where θ is the angle at the centre in degrees, d is the diameter and r is the radius.
For a semicircle (θ = 180°): arc length = (180/360) × πd = πd/2 = πr
For a quarter circle (θ = 90°): arc length = (90/360) × πd = πd/4 = πr/2
How do you find the perimeter of a shape containing a semicircle?
Worked example 1: A shape consists of a rectangle 10 cm by 6 cm with a semicircle attached to one of the 6 cm ends. Find the perimeter.
The semicircle has diameter 6 cm (radius 3 cm).
Identify every edge on the boundary:
- Long side (top): 10 cm
- Long side (bottom): 10 cm
- Short side (left): 6 cm
- Semicircular arc (right): (180/360) × π × 6 = 3π cm
Note: The diameter of the semicircle (the 6 cm straight edge on the right) is inside the shape — it is not part of the perimeter. Only the curved arc forms the boundary on that side.
Total perimeter = 10 + 10 + 6 + 3π = 26 + 3π ≈ 35.4 cm (to 1 d.p.)
How do you find the perimeter of a sector?
A sector is a "pie slice" shape bounded by two radii and an arc. Its perimeter includes two straight sides (the radii) plus the arc.
Perimeter of sector = 2r + arc length = 2r + (θ/360) × 2πr
Worked example 2: Find the perimeter of a sector with radius 8 cm and angle 135°.
- Arc length = (135/360) × 2π × 8 = (3/8) × 16π = 6π cm
- Two radii: 2 × 8 = 16 cm
- Total perimeter = 16 + 6π = 16 + 6π ≈ 34.9 cm
How do you handle a shape where an arc is cut out?
Sometimes a shape has a curved section removed from its boundary — for example, a square with a quarter-circle notch cut from one corner.
Worked example 3: A square of side 12 cm has a quarter-circle of radius 5 cm cut from one corner. Find the perimeter.
Boundary edges:
- Two full sides of the square: 12 + 12 = 24 cm
- Two sides that are shortened: (12 − 5) + (12 − 5) = 7 + 7 = 14 cm
- The quarter-circle arc (this replaces the cut corner): (90/360) × 2π × 5 = (1/4) × 10π = 2.5π cm
Total perimeter = 24 + 14 + 2.5π = 38 + 2.5π ≈ 45.9 cm
The key is to identify which edges from the original shape have been removed and replace them with the arc.
How do you deal with shapes combining multiple arcs?
Worked example 4: A running track consists of two straight sections each 100 m long and two semicircular ends each of diameter 80 m. Find the total perimeter.
- Two straight sections: 2 × 100 = 200 m
- Two semicircular arcs: each has arc length = (180/360) × π × 80 = 40π m; total = 2 × 40π = 80π m
- Total = 200 + 80π = 200 + 80π ≈ 451.3 m
What are the most common errors with perimeter and arcs?
| Error | What goes wrong | Fix |
|---|---|---|
| Including the diameter as part of the perimeter | Counting a chord or diameter that is inside the shape | Only external boundary edges count |
| Using diameter instead of radius in 2πr | Doubling the radius twice | Check: if using 2πr, r is radius; if using πd, d is diameter |
| Rounding π early | Small rounding error compounded by addition | Keep as multiples of π until the very last step |
| Wrong angle | Using the reflex angle instead of the interior angle | Read the angle from the diagram carefully; use the angle marked at the centre |
| Forgetting the straight sides of a sector | Calculating only the arc | Add both radii to the arc for the sector's full perimeter |
Frequently asked questions
How do I know which sides form the perimeter of a composite shape?
The perimeter is the distance around the outside of the shape. Imagine an ant walking around the outside — every edge it crosses is part of the perimeter. Any line that is internal (shared between two joined parts) is not on the perimeter. Sketch the shape and mark each external edge before calculating.
Should I give arc length answers in terms of π or as a decimal?
Exam mark schemes accept both, but leaving answers in terms of π is exact (e.g. 6π cm) while decimal answers are approximate (18.85 cm). Unless the question says "give your answer to 1 decimal place" or similar, an exact answer is often preferred. Check the question's instruction.
What is the difference between arc length and sector area?
Arc length is a one-dimensional measurement — the length of the curved edge. Sector area is a two-dimensional measurement — the area of the pie-slice region. Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr². Both formulae use the same fraction of the circle; the difference is whether you multiply by 2πr (circumference) or πr² (area).
Can the perimeter formula be used for a full circle?
Yes — a full circle has θ = 360°, so arc length = (360/360) × 2πr = 2πr, which is just the circumference. The perimeter of a circle is its circumference: C = 2πr = πd. The arc length formula is the general version that works for any angle.
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