Geometry and Measurement
Geometry and Measurement in the Pancyprian Core Mathematics paper is largely deductive: you are expected to name the theorem you use and to move correctly between lengths, areas and volumes of similar figures. Mensuration formulas are rarely supplied, so they must be known together with their constants ½, ⅓ and π.
Triangles and similarity
- Pythagoras' theorem, and Thales' theorem on parallel lines cutting proportional segments.
- Two triangles are similar when two angles are equal (AA); corresponding sides are then in a fixed ratio k.
- For similar figures: lengths scale by k, areas by k², volumes by k³.
The circle
- The angle at the centre is twice the inscribed angle standing on the same arc; the angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral are supplementary.
- A tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal in length.
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Mensuration
- Area of a triangle = ½ab sin C; sector area = ½r²θ with θ in radians; arc length s = rθ.
- Volume: prism and cylinder = base area × height; pyramid and cone = ⅓ × base area × height; sphere = (4/3)πr³.
- Surface area: sphere 4πr²; cone πrl, where l is the slant height and not the perpendicular height.
Worked example
Two similar solid cones have volumes 27 cm³ and 64 cm³. The total surface area of the smaller is 36 cm². Find the total surface area of the larger. The volume ratio is 27 : 64, so the ratio of lengths is k = ∛(27/64) = 3/4, giving larger : smaller = 4/3. Areas scale by k² = 16/9, so the larger surface area is 36 × 16/9 = 64 cm².
Traps that cost marks
- Applying the length ratio directly to volumes instead of cubing it, or to areas instead of squaring.
- Substituting the diameter for the radius in πr², 4πr² and (4/3)πr³.
- Omitting the factor ⅓ for cones and pyramids.
- Treating a printed sketch as drawn to scale and reading a length from it.