ShieldsGeometry
Spatial GeometryPlane GeometryCoordinates & VectorsApplied & ExamExam papers

Shields Geometry | Geometry lessons, formulas & exam practice

Clear geometry notes for Hong Kong students — from junior secondary foundations through HKDSE-style timed practice. Each lesson includes formulas, a worked example and short drills.

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1Pyramid volume: identify the base and perpendicular heightSpatial Geometry · Ms Tam · V = (1/3) × base area × perpendicular height2Distance from a point to a plane using volumeSpatial Geometry · Ms Tam · V = (1/3) × base area × height3Finding the angle between a line and a planeSpatial Geometry · Ms Tam · θ = angle between the line and its projection4Circle tangents: the radius is perpendicularPlane Geometry · Ms Ho · tangent ⟂ radius at the point of contact5Writing the equation of a plane in 3DSpatial Geometry · Mr Wong · Ax + By + Cz + D = 06Right-triangle lengths: Pythagoras and altitudePlane Geometry · Ms Ho · a² = b² + c²7Geometry formula sheet with quick checksApplied & Exam · Mr Wong · triangle area = (1/2)bh8A timed geometry plan for HKDSE-style papersApplied & Exam · Mr Wong · accuracy first, then speed9Cylinder volume and curved surface areaSpatial Geometry · Ms Tam · V = πr²h10Similar triangles: matching sides in proportionPlane Geometry · Ms Ho · AB/DE = BC/EF = AC/DF11Chord theorem: perpendicular from the centrePlane Geometry · Ms Ho · OM ⟂ AB ⇒ AM = MB12Gradient of a line through two pointsCoordinates & Vectors · Mr Wong · m = (y₂ − y₁) / (x₂ − x₁)13Section formula with position vectorsCoordinates & Vectors · Mr Wong · If P divides AB in m:n, then OP = (n·OA + m·OB) / (m + n)14Cone volume when slant height is givenSpatial Geometry · Ms Tam · l² = r² + h²15Isosceles triangles and equal base anglesPlane Geometry · Ms Ho · AB = AC ⇒ ∠ABC = ∠ACB16Reflection across the axes and y = xCoordinates & Vectors · Mr Wong · reflection in x-axis: (x, y) → (x, −y)17Parallelogram area from two vectorsCoordinates & Vectors · Mr Wong · area = |x₁y₂ − x₂y₁| for a = (x₁, y₁), b = (x₂, y₂)18Bearings and a right-triangle journeyApplied & Exam · Ms Tam · bearing measured clockwise from north19Alternate segment theorem with a short proofPlane Geometry · Ms Ho · ∠ between tangent and chord = angle in alternate segment20Map scales and real distancesApplied & Exam · Ms Tam · real length = map length × scale factor21Prism volume from a cross-sectionSpatial Geometry · Ms Tam · V = cross-sectional area × length22Sphere volume without the fear of πSpatial Geometry · Ms Tam · V = (4/3)πr³23Space diagonal of a cubeSpatial Geometry · Mr Wong · face diagonal = a√224Volume of a triangular pyramidSpatial Geometry · Ms Tam · V = (1/3) × triangle area × height25Frustum of a pyramid: the cut-off topSpatial Geometry · Ms Tam · V = (1/3)h(A₁ + A₂ + √(A₁A₂))26Angle between two planesSpatial Geometry · Ms Ho · cos φ = |n₁ · n₂| / (|n₁||n₂|)27Shortest distance between skew linesSpatial Geometry · Mr Wong · d = |(P₂−P₁) · n| / |n|28Largest cylinder inside a sphereSpatial Geometry · Ms Tam · r² + (h/2)² = R²29Cube nets and surface areaSpatial Geometry · Ms Ho · surface area = 6a²30Finding height from a lateral edgeSpatial Geometry · Ms Tam · h = √(ℓ² − d²)31Plane determined by three pointsSpatial Geometry · Mr Wong · n = AB × AC32Hemisphere volume and curved surfaceSpatial Geometry · Ms Tam · V = (2/3)πr³33Space diagonal of a rectangular boxSpatial Geometry · Mr Wong · d = √(a² + b² + c²)34Volume of a hollow pipeSpatial Geometry · Ms Tam · V = π(R² − r²)h35True height from an oblique sketchSpatial Geometry · Ms Ho · mark ⊥ before measuring height36Cone: from slant height to perpendicular heightSpatial Geometry · Ms Tam · ℓ² = r² + h²37Composite solids: add or subtract volumesSpatial Geometry · Ms Tam · V_total = V₁ ± V₂ ± …38Circular cross-section of a sphereSpatial Geometry · Mr Wong · ρ = √(R² − d²)39Surface area of a square-based pyramidSpatial Geometry · Ms Ho · total SA = a² + 2aℓ40Converse of Pythagoras: proving a right anglePlane Geometry · Ms Ho · a² + b² = c² ⇒ angle opposite c is right41Heron’s formula when height is hiddenPlane Geometry · Ms Ho · s = (a+b+c)/242Exterior angle of a trianglePlane Geometry · Ms Ho · exterior = sum of remote interiors43Parallel lines and a transversalPlane Geometry · Ms Ho · alt = alt44Angle at the centre is twice the angle at the circumferencePlane Geometry · Ms Ho · ∠ at centre = 2 × ∠ at circumference45Cyclic quadrilateral: opposite anglesPlane Geometry · Ms Ho · ∠A + ∠C = 180°46AA similarity in two stepsPlane Geometry · Ms Ho · AA ⇒ similar47SAS congruence without mixing the anglePlane Geometry · Ms Ho · SAS, ASA, SSS, RHS48Tangent–secant length theoremPlane Geometry · Ms Ho · PT² = PA × PB49Height and area of an equilateral trianglePlane Geometry · Ms Tam · h = (√3/2)a50Midpoint theorem in a trianglePlane Geometry · Ms Ho · DE ∥ BC and DE = BC/2