lesson
Trace the outside, then find the area
1 · Learn
A composite shape is made from simpler shapes. Before calculating, mark which lengths are given and find any missing lengths by addition or subtraction. Area measures the covered region; perimeter measures the OUTSIDE boundary. Internal joins are not part of the perimeter.
The L-shape shown is an 8 cm by 6 cm rectangle with a 3 cm by 2 cm corner removed. Its area is 8×6 - 3×2 = 42 cm². The remaining top length is 8-3 = 5 cm and the right vertical length is 6-2 = 4 cm. Trace the six outside edges: 5+2+3+4+8+6 = 28 cm.
Here the perimeter happens to equal the original rectangle's perimeter. The removed corner takes away lengths 3 and 2, but the new inside notch adds lengths 3 and 2. The area changes, so equal perimeters do not imply equal areas. This argument is for this corner cut, not every possible hole or cut-out.
For a curved example, attach a semicircle to one short side of an 8 cm by 6 cm rectangle. The shared diameter is 6 cm, so the radius is 3 cm. Area = rectangle + half-circle = 48 + (9/2)π cm². Keep this exact form unless a decimal approximation is requested.
For the same curved shape, the outside includes two 8 cm sides, ONE straight 6 cm side and the curved semicircle arc. Perimeter = 8+8+6+3π = 22+3π cm. The dashed shared diameter in the diagram is internal: including it would count a line that is not on the outside. Check units as well as the formula.
2 · Worked example
Remove a 4 cm by 3 cm top-right corner from a 10 cm by 7 cm rectangle. Area = 70-12 = 58 cm². The six boundary lengths are 6,3,4,4,10,7 cm, totalling 34 cm.
3 · Your turn
A semicircle is attached along one 4 cm side of a 10 cm by 4 cm rectangle. Find the exact area and outside perimeter. Which line must not be counted in the perimeter?
Check your answer
Radius 2 cm. Area 40+2π cm². Outside perimeter 10+10+4+2π = 24+2π cm. Do not count the shared 4 cm diameter.
4 · Apply your learning
Sketch both supplied examples on paper or the working pad. Colour the area, then use a different colour to trace only the outside. Label the missing straight lengths and the semicircle radius before using a formula.
Adult guidance and safety
A sketch supports reasoning; use the given measurements rather than measuring a screen. Accept equivalent exact forms such as 9π/2 and (9/2)π. Provide squared paper if a pupil needs to build the area model first.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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