ShieldsGeometry
Spatial GeometryPlane GeometryCoordinates & VectorsApplied & ExamExam papers

Geometry lessons

100 lessons

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/251Basic proportionality (Thales) theoremPlane Geometry · Ms Ho · AD/DB = AE/EC52Angles in the same segmentPlane Geometry · Ms Ho · angles in the same segment are equal53Rhombus diagonals are perpendicular bisectorsPlane Geometry · Ms Ho · d₁² + d₂² = 4a²54Area of a trapeziumPlane Geometry · Ms Tam · area = ½(a+b)h55Sine rule for a non-right trianglePlane Geometry · Mr Wong · a/sin A = b/sin B = c/sin C56Cosine rule to find a sidePlane Geometry · Mr Wong · c² = a² + b² − 2ab cos C57Angle bisector theoremPlane Geometry · Ms Ho · BD/DC = AB/AC58Two tangents from an external pointPlane Geometry · Ms Ho · PT = QT59Distance between two pointsCoordinates & Vectors · Mr Wong · d = √[(x₂−x₁)²+(y₂−y₁)²]60Midpoint and section formula on a lineCoordinates & Vectors · Mr Wong · M = ((x₁+x₂)/2,(y₁+y₂)/2)61Equation of a line through two pointsCoordinates & Vectors · Mr Wong · m = (y₂−y₁)/(x₂−x₁)62Parallel and perpendicular gradientsCoordinates & Vectors · Mr Wong · parallel: m₁=m₂63Area of a triangle from coordinatesCoordinates & Vectors · Mr Wong · area = ½|det form|64Circle equation from centre and radiusCoordinates & Vectors · Mr Wong · (x−h)²+(y−k)²=r²65Intersection of a line and a circleCoordinates & Vectors · Mr Wong · discriminant >0 two points66Vector addition as a parallelogramCoordinates & Vectors · Mr Wong · a + b = diagonal67Position vectors and midpointsCoordinates & Vectors · Mr Wong · OM = (a+b)/268Rotation by 90° about the originCoordinates & Vectors · Ms Ho · 90° ACW: (x,y)→(−y,x)69Enlargement from a centreCoordinates & Vectors · Ms Ho · X′ = C + k(X − C)70Locus: perpendicular bisectorCoordinates & Vectors · Mr Wong · |PA|=|PB| ⇒ P on perp. bisector of AB71Proving collinearity with vectorsCoordinates & Vectors · Mr Wong · AB = k AC72Dot product and right anglesCoordinates & Vectors · Mr Wong · a · b = |a||b|cos θ73Parametric equation of a lineCoordinates & Vectors · Mr Wong · r = a + t d74Translation by a vectorCoordinates & Vectors · Ms Ho · (x,y) → (x+a,y+b)75Coordinate proof that a quadrilateral is a parallelogramCoordinates & Vectors · Mr Wong · midpoint AC = midpoint BD76Shortest path via a mirror lineCoordinates & Vectors · Ms Ho · reflect → straight line → intersect mirror77Gradient as tan of the inclinationCoordinates & Vectors · Mr Wong · m = tan θ78Dividing a segment in a vector ratioCoordinates & Vectors · Mr Wong · p = (n a + m b)/(m+n)79Command words: find, show, explain, henceApplied & Exam · Mr Wong · read the verb before calculating80Annotating the exam diagram in the first minuteApplied & Exam · Ms Tam · given → mark → plan → calculate81A 30-second unit and scale checkApplied & Exam · Mr Wong · length / area / volume → unit82Map scales and area factorsApplied & Exam · Ms Tam · area factor = (length factor)²83Back bearings and reverse journeysApplied & Exam · Mr Wong · back bearing = bearing ± 180° (mod 360°)84Percentage error in a measured lengthApplied & Exam · Ms Tam · % error = (|approx − true|/true)×100%85Two-pass timing for a mixed geometry paperApplied & Exam · Mr Wong · pass 1 easy marks → pass 2 constructions86Roof pitch as a right triangleApplied & Exam · Ms Tam · tan θ = rise/run87Ladder against a wallApplied & Exam · Ms Ho · ladder² = height² + base²88Heights from shadows with similar trianglesApplied & Exam · Ms Tam · h₁/s₁ = h₂/s₂89Water tank problems with consistent unitsApplied & Exam · Ms Tam · 1 m³ = 1000 L90Keeping a one-week geometry error logApplied & Exam · Mr Wong · classify → correct → retest91Choosing formulas by the quantity namedApplied & Exam · Mr Wong · target quantity → formula family92Exact trig values you should not calculatorApplied & Exam · Ms Ho · sin30=1/293Paint coverage from surface areaApplied & Exam · Ms Tam · litres = area / coverage94Layout that earns method marksApplied & Exam · Mr Wong · formula → substitute → simplify → conclude95Compound shapes: add and subtract areasApplied & Exam · Ms Ho · A = A₁ + A₂ − overlap96Spider-and-fly shortest path on wallsApplied & Exam · Mr Wong · shortest path = straight line on a suitable net97Rounding lengths without wrecking the next stepApplied & Exam · Mr Wong · round at the end98Calculator degree mode and inverse trigApplied & Exam · Ms Ho · θ = tan⁻¹(opp/adj) in DEG mode99Arc length and sector areaPlane Geometry · Ms Ho · arc = 2πr·(θ/360°)100Estimating before calculatingApplied & Exam · Ms Tam · estimate → calculate → compare
Geometry lessons — Shields Geometry