Model before calculating
A composite answer begins with a ledger: required region or boundary, component shapes, operations, repetition count and units.
Learning goals
- Decompose a composite diagram into named measurable regions and boundaries.
- Combine repeated or concentric circular measures without double counting.
- Model contextual swept regions and complete multi-part targets.
Concept lesson
Circular component measures
Recall circle area, circumference, arc length, sector area and sector perimeter; identify whether the context asks for covered region or boundary material.
Quantity and unit check
Boundary uses linear units, region uses square units, and cost equals an area or length multiplied by its matching rate.
Region and boundary ledgers
For area, list non-overlapping filled components with plus or minus signs. For perimeter, trace only the requested exposed boundary and list each arc or straight segment once.
Repetition and scale
Equal ribs divide 360 degrees equally; multiple non-overlapping wipers multiply one swept sector; repeated designs multiply one component only after overlap is excluded.
Grazing at a field corner
A rope tied at a square-field corner sweeps a 90-degree sector if its length stays within the field. For increased grazing area, subtract the old quadrant area from the new one.
Two non-overlapping wipers
Calculate one blade's sector from its length and sweep angle, then multiply by two only because the question states that the swept regions do not overlap.
Trace the true outer boundary
Trace a fan or arc-built design continuously, mark internal construction lines as excluded, and sum only exposed arcs and required straight supports.
Count equal sectors and repetitions
Convert rib or design count to a central angle, calculate one component, record the repetition factor, and verify that the total angular allocation is 360 degrees.
Complete every requested output
Create an output checklist for area, increase, arc, perimeter or cost; attach the correct unit to each intermediate and final quantity.
Composite design reasoning chain
Build a signed component ledger for a shaded design, a separate boundary ledger if needed, apply repetition exactly once, then convert the result to cost or change where requested.
Marks follow the modelling chain
Show the angle or repetition conversion, component formulae, signed combination and requested final unit; a bare number hides the modelling evidence.
Composite-circle mastery
Solve one swept-region and one arc-built design problem using explicit region, repetition, boundary and unit ledgers.