Mensuration and exact answers
Break a compound shape into familiar pieces and preserve exact values.
What you will learn
- Find areas and volumes of shapes with parts removed.
- Keep π symbolic when an exact answer is required.
Plan the geometry before calculating
Identify which regions are included and which are removed. For a trapezium, average the parallel sides and multiply by the perpendicular height. For a semicircle, halve the circle area. A diameter must be halved before using a formula involving radius.
An area calculation assumes that the removed region lies within the original shape. Read placement descriptions carefully rather than assuming that any two areas can be subtracted.
Use cross-sections for constant solids
The volume of a prism is its cross-sectional area multiplied by its length. A cylinder with a coaxial cylindrical hole has an annular cross-section. Subtract the inner circle area from the outer circle area, then multiply by the common height.
Exact answers containing π should stay in terms of π. A decimal approximation such as 3.14 changes an exact expression into an approximation.
Worked example
A cylinder has radius 5 cm and height 4 cm. A coaxial hole of radius 2 cm passes through it. Find the exact remaining volume.
Show the worked solution
- The cross-sectional area is π(25−4)=21π cm².
- The shape has the same cross-section throughout its height.
- Multiply by 4 cm.
Answer 84π cm³
Common mistakes
- Subtract squared radii, not the square of their difference.
- Area uses square units; volume uses cubic units.
What can you explain now?
Find the exact area of a semicircle with diameter 10 cm.
Compare with the explanation
25π/2 cm²
The radius is 5 cm, so half of π×5² is 25π/2.
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