Compound shapes
Section: Mensuration | Syllabus: Cambridge IGCSE Mathematics (0580)
Area and Perimeter of Compound Shapes
A compound shape is made up of two or more simple shapes joined together. Its area and perimeter can be found by splitting it into simple shapes first.
- Split the compound shape into simple shapes (rectangles, triangles, etc.) whose areas can be calculated individually
- Add the areas of the separate parts together to find the total area
- For the perimeter, add up the length of every outer edge - be careful not to include any internal lines used only to split the shape
- Sometimes it is easier to find the area of a larger simple shape and subtract a smaller piece removed from it, rather than adding pieces together
Worked Example: Finding the Area of an L-Shaped Figure
- Question: An L-shaped figure has a top edge of 12 cm and a left edge of 9 cm. A rectangular notch of width 5 cm and height 4 cm is cut from the bottom-right corner. Find the area of the shape.
- Step 1: Split into two rectangles: a top strip 12×(9-4)=125, and a lower strip (12-5)4=74
- Step 2: Rectangle 1 area: 125=60
- Step 3: Rectangle 2 area: 74=28
- Step 4: Total area: 60+28=88
- Answer: 88 cm²
Common Mistakes
MistakeIncluding an internal splitting line as part of the perimeter, adding extra length that isn't on the outer boundary
Fixonly add up the lengths of the shape's actual outer edges - any line drawn purely to split the shape into rectangles should be excluded from the perimeter
MistakeMiscalculating a missing side length needed to split the shape, especially where two given lengths must be subtracted to find it
Fixsketch the split shapes clearly and label every side, working out any missing lengths from the given ones before calculating areas
Area of Compound Shapes Involving Circles
Compound shapes often combine circles (or parts of circles) with other shapes, or involve one circle cut out from inside another.
- When a smaller shape is removed from within a larger one, the area of the remaining (shaded) region = larger area − smaller area
- Always find the radius of each circle first (halving the diameter if necessary) before applying the circle area formula
- Leave answers in terms of π unless told to round to a decimal
Worked Example: Finding a Shaded Area Between Two Circles
- Question: A large circle has diameter 24 cm. A smaller circle with diameter 10 cm lies entirely inside it. Find the shaded area (inside the large circle but outside the small circle), in terms of π.
- Step 1: Large circle radius = 12 cm, area: π12^2=144π
- Step 2: Small circle radius = 5 cm, area: π5^2=25π
- Step 3: Shaded area: 144π-25π=119π
- Answer: 119π cm²
Common Mistakes
MistakeAdding the two circle areas together instead of subtracting, when the smaller shape has been removed from the larger one
Fixread the diagram carefully - if a region is unshaded (a "hole"), its area must be subtracted, not added
MistakeUsing the given diameters directly in the area formula πr², instead of first halving them to find each radius
Fixalways check whether the given measurement is a radius or a diameter before substituting into πr²
Estimating the Area of Irregular Shapes
Some shapes have curved or irregular edges that don't match any standard formula. Their area can be estimated by counting squares on a grid.
- Count every whole square that lies completely inside the shape first
- For each square only partly inside the shape, estimate whether it counts as roughly half a square (or a full square, if more than half is covered), and add these estimates to the whole-square count
- Since this method only gives an estimate, a range of close answers is usually accepted, not just one exact value
Worked Example: Estimating an Area by Counting Squares
- Question: A shape is drawn on a 1 cm² grid. It fully covers 14 whole squares, and has 10 part-squares around its curved edge, each roughly half-covered. Estimate the area of the shape.
- Step 1: Whole squares = 14
- Step 2: Part-squares: 100.5=5
- Step 3: Total estimate: 14+5=19
- Answer: approximately 19 cm²
Common Mistakes
MistakeOnly counting the whole squares and ignoring the part-squares around curved edges entirely
Fixpart-squares must still be estimated and included - ignoring them gives a significant underestimate
MistakeExpecting (or insisting on) one single exact answer for an estimated area
Fixcounting squares only ever gives an estimate - a small range of values around the true area is normally accepted
Volume and Surface Area of Compound Solids
A compound solid is made by joining two or more simple 3D solids together, such as a cone sitting on top of a cylinder. Its volume and surface area are found using the same split-and-combine approach as for 2D compound shapes.
- Split the compound solid into simple solids whose volume formulas are known, then add the volumes together
- For surface area, only include the faces that are actually on the outside of the finished compound solid - any face where two solids join together is hidden inside, and must be left out
- Check carefully which radius, height or other measurement belongs to which part of the compound solid
Worked Example: Finding the Volume of a Compound Solid
- Question: A solid is made from a cylinder of radius 4 cm and height 10 cm, with a cone of the same radius and height 6 cm attached on top. Find the total volume of the solid, in terms of π.
- Step 1: Volume of cylinder: π4^210=160π
- Step 2: Volume of cone: (1/3)×π4^26=32π
- Step 3: Total volume: 160π+32π=192π
- Answer: 192π cm³
Common Mistakes
MistakeIncluding the circular face where the cone and cylinder join when calculating the total surface area
Fixthe joining face is hidden inside the compound solid, not on its outer surface - it must be left out of a surface area calculation
MistakeUsing the wrong height for one part of the compound solid, e.g. using the cylinder's height for the cone's volume
Fixidentify each solid's own measurements separately before substituting into its formula, especially when different parts share the same radius
Compound Shapes Involving Algebraic Expressions
Extended Only
Some compound shape problems use algebraic expressions instead of numbers for the side lengths, requiring the areas of the split shapes to be expanded and simplified.
- Split the shape into simple rectangles exactly as with numerical compound shapes, but keep every length as an algebraic expression
- Multiply out (expand) each rectangle's area expression carefully, then collect like terms when combining them
- Give the final answer in its simplest form, factorising if asked to
Worked Example: Finding an Algebraic Expression for a Compound Area
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