Compound areas and outside perimeters
Split or subtract areas, then trace only the exposed boundary.
Core and Extended.
Before you begin
- Use rectangle, triangle and circle measures.
- Identify sector arcs and straight boundaries.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When two shapes share a joining edge, is that edge on the outside boundary?
What you will learn
- Find area of joined shapes, cut-outs and rings.
- Find perimeter without counting internal edges.
Partition without overlaps or gaps
A compound area can be split into familiar non-overlapping shapes, or found by subtracting a missing region from a larger enclosing shape. A 10 by 8 cm rectangle with a 4 by 3 cm corner removed has area 80 − 12 = 68 cm². Show the split or cut-out so each region is counted once.
A circular ring of outer radius R and inner radius r has area π(R² − r²). It is not π(R − r)²: the ring's thickness is not a radius of a replacement circle. If pieces overlap, adding their whole areas double-counts the overlap; subtract that shared region or partition it separately.
Area bookkeeping does not give perimeter
Perimeter follows the exposed outer boundary, plus any inner boundary explicitly requested. Joining a semicircle to a rectangle along its diameter removes that shared diameter from the outside. A rectangle 8 cm long and 6 cm wide with a radius-3 semicircle on one short end has perimeter 8 + 8 + 6 + 3π = 22 + 3π cm.
Removing a corner can leave the same total perimeter: in the 10 by 8 rectangle above, removed boundary lengths 4 and 3 are replaced by notch edges of those same lengths, so perimeter remains 36 cm. A hole creates a new inner boundary, while a filled join hides an old boundary. Trace the specified edges instead of subtracting a cut-out's whole perimeter.
Pause and explain
A circle ring has outer radius 5 cm and inner radius 3 cm. Find its area.
Worked example
A 10 by 8 cm rectangle has a 4 by 3 cm rectangular corner removed. Find the remaining area and perimeter.
Show the worked solution
- Remaining area is 10 × 8 − 4 × 3 = 68 cm².
- Two removed outside lengths total seven centimetres; two new notch edges also total seven.
- The outside perimeter stays 2(10 + 8) = 36 cm.
Answer Area 68 cm²; perimeter 36 cm.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A rectangle 8 by 6 cm has a radius-3 cm semicircle joined along one 6 cm end. Find total area and outside perimeter.
Give me a hint
Add the filled areas but omit the shared diameter from perimeter.
Compare my reasoning
- Area is 48 + (1/2)π × 9 = 48 + 9π/2 cm².
- Three exposed rectangle sides total 8 + 8 + 6 = 22 cm.
- The semicircle arc adds 3π cm, giving perimeter 22 + 3π cm.
Area (48 + 9π/2) cm²; perimeter (22 + 3π) cm.
Look for these in your work
- Counted each region once.
- Omitted the joined diameter.
A ring has outer radius 5 cm and inner radius 3 cm. Find area and the combined length of both circular edges.
Give me a hint
Both edges count only because the question explicitly asks for both.
Compare my reasoning
- Area is π(25 − 9) = 16π cm².
- The outer edge is 10π cm and inner edge is 6π cm.
- Combined edge length is 16π cm, which has length units despite the same coefficient.
Area 16π cm²; combined edge length 16π cm.
Look for these in your work
- Subtracted the hole area.
- Included both specified boundaries.
A stadium-shaped track surrounds a 10 by 6 m rectangle with a semicircle at each short end. Find enclosed area and outside perimeter.
Give me a hint
Two semicircles make one circle of radius three metres.
Compare my reasoning
- Area is 60 + 9π m².
- The two straight exposed sides total 20 m.
- The two curved ends total 6π m, giving perimeter 20 + 6π m.
Area (60 + 9π) m²; perimeter (20 + 6π) m.
Look for these in your work
- Combined the semicircles correctly.
- Excluded both internal diameters.
Common mistakes
- Counting internal joins in outside perimeter.
- Adding a hole's area.
- Squaring the difference of ring radii instead of subtracting their squares.
What can you explain now?
Can a removed corner leave perimeter unchanged?
Compare with the explanation
Yes.
New notch edges can replace removed boundary lengths exactly, so trace the boundary rather than assuming subtraction.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, draw a joined rectangle and semicircle. Colour area pieces and trace exposed edges separately.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources