The perimeter of a ring sector is the sum of its outer arc, its inner arc and its two straight edges. Each straight edge spans the difference between the two radii. The inner arc counts because it forms part of the boundary of the remaining shape.
This 2027 SEC G3 Mathematics practice applies arc length to a piece cut from a ring. Sketch the outline first so you can see exactly which lengths the question asks you to add.
- Mark the inner and outer radii.
- Calculate each arc using the same central angle.
- Find the length of one straight edge.
- Add the four boundary pieces.
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Trace the outline before calculating
Picture two concentric circles. Draw two rays from their common centre, and shade the region between the two circles and the two rays. Start at a corner on the outside. Trace the outer arc, cross to the inner circle along one straight edge, follow the inner arc, then cross back along the other straight edge.
The centre lies outside the shaded piece. Each straight edge starts at the inner circle and ends at the outer circle. If the outer radius is R and the inner radius is r, that straight length is R − r.
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Worked example | A 90° ring sector
The outer radius is 10 cm and the inner radius is 6 cm. A central angle of 90° makes each arc one quarter of its own circle. The outer arc measures ¼ × 2π × 10 = 5π cm. The inner arc measures ¼ × 2π × 6 = 3π cm.
Each straight edge is 10 − 6 = 4 cm. The perimeter is 5π + 3π + 4 + 4 = 8π + 8 cm, or about 33.1 cm. Keeping π through the working avoids rounding each arc separately.
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Change the angle and predict which lengths change
If the central angle becomes 180° with the same radii, both arcs double. The two straight edges stay at 4 cm each. The new perimeter is 16π + 8 cm. This check helps explain why doubling the angle does not double the entire perimeter.
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Frequently asked questions
When would I subtract the inner part?
Subtract the inner sector area from the outer sector area to find the shaded area. For the perimeter, trace both arcs because both are boundary edges.
How can a parent check the method?
Ask the student to point to four pieces on a sketch and match each piece to one term in the calculation.