Geometry and Mensuration
Geometry and Mensuration covers the angle properties of lines, triangles and polygons, circle theorems, coordinate geometry, bearings, and the mensuration of plane shapes and solids (prisms, cylinders, cones, spheres, pyramids and frustums). NECO marks reasoning: a correct value with a missing or wrong theorem stated can lose the grade.
Angle facts
- Angles on a straight line sum to 180°, at a point to 360°; vertically opposite angles are equal.
- Parallel lines: alternate and corresponding angles are equal, co-interior angles sum to 180°.
- Any polygon: exterior angles sum to 360°; interior angles sum to (n − 2) × 180°.
Circle theorems
- Angle at the centre = 2 × angle at the circumference.
- Angle in a semicircle = 90°.
- Angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.
- A tangent is perpendicular to the radius at the point of contact; alternate segment theorem.
Coordinate geometry
Gradient m = (y₂ − y₁)/(x₂ − x₁); distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2); line: y − y₁ = m(x − x₁). Parallel lines share m; perpendicular lines satisfy m₁m₂ = −1.
Mensuration
- Cone: l = √(r² + h²); curved surface πrl; total surface πr(l + r); volume ⅓πr²h.
- Cylinder: curved 2πrh; total 2πr(r + h); volume πr²h.
- Sphere: surface 4πr²; volume (4/3)πr³.
Worked example
A cone has radius 7 cm and height 24 cm. Find its total surface area and volume, taking π = 22/7. Slant height l = √(7² + 24²) = √625 = 25 cm. Total surface area = πr(l + r) = (22/7)(7)(25 + 7) = 22 × 32 = 704 cm². Volume = ⅓πr²h = ⅓ × (22/7) × 49 × 24 = 22 × 7 × 8 = 1232 cm³.
Traps
- Using the perpendicular height in πrl instead of the slant height.
- Omitting the base when a question asks for total surface area.
- Treating a diagram as drawn to scale and reading an angle that is not given.