Area of composite shapes
Account for each part of the required region exactly once
\[A=A_{\mathrm{whole}}-A_{\mathrm{removed}}\]
- Recover missing dimensions For a 9 m full span and a matching 4 m partial span, subtract to find the missing length \(\displaystyle 9-4=5\text{ m}\)
- Add non-overlapping pieces For a rectangle of width w and height h with an attached semicircle of radius r, add the two non-overlapping areas \(\displaystyle A=wh+\frac12\pi r^2\)
- Subtract holes and cut-outs From a w by h rectangle, subtract a c by d cut-out \(\displaystyle A=wh-cd\)
- Use the required sector For a sector, multiply the full circle area by the angle’s fraction of a full turn \(\displaystyle A_{\mathrm{sector}}=\frac{\theta}{360}\pi r^2\)
Worked example
A 15 cm by 10 cm rectangle has a 5 cm by 4 cm corner removed. Find the remaining area
\[\begin{gathered}w=15\text{ cm},\quad h=10\text{ cm}\\5\text{ cm}\times4\text{ cm cutout}\end{gathered}\]
- \(\displaystyle \begin{gathered}A=15\times10-5\times4\\=150-20=130\text{ cm}^2\end{gathered}\)Subtract the cut-out area from the enclosing rectangle
Common mistakes
Adding a cut-out
✕\(\displaystyle \begin{gathered}A=15\times10+5\times4\\=170\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}A=15\times10-5\times4\\=130\text{ cm}^2\end{gathered}\)
A removed region is subtracted from the enclosing area rather than addedCounting an overlapping region twice
✕\(\displaystyle A=A_1+A_2\)✓\(\displaystyle A=A_1+A_2-A_{\mathrm{overlap}}\)
If two chosen regions overlap, their sum counts the overlap twice. Subtract the overlap once, or choose non-overlapping piecesUsing a semicircle diameter as its radius
✕\(\displaystyle A_{\mathrm{semi}}=\frac12\pi(8)^2\)✓\(\displaystyle A_{\mathrm{semi}}=\frac12\pi(4)^2\)
A semicircle of diameter 8 m has radius 4 m. Halve the diameter before squaringDifficulty levels
introSubtract a rectangular corner from a rectangle
standardDecompose T-shaped and stepped regions
challengeCombine rectangles, triangles and circular pieces, rounding only at the end
examRecognise composite area from distinct same-unit options
Questions about this topic
When should areas be added or subtracted?+
Add non-overlapping regions that are part of the shape. Subtract a hole or cut-out from an enclosing region
Do shared edges contribute area?+
An edge has no area. It separates the component regions but is not an extra region to add
How can I check a subtraction answer?+
The remaining area plus the removed area must equal the enclosing area
Is every semicircle measurement a radius?+
No. A straight edge across the whole semicircle is its diameter. Halve it before using the circle area formula
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)