IB›IB Maths: Analysis and Approaches HL›Mind maps3.1 Three-dimensional geometryIB Maths: Analysis and Approaches HL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsDistance, midpointd=(Δx)2+(Δy)2+(Δz)2d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}d=(Δx)2+(Δy)2+(Δz)2Midpoint: average each coordinateA(2,−1,5)A(2,-1,5)A(2,−1,5), B(8,3,−7)B(8,3,-7)B(8,3,−7): AB=14AB=14AB=14Unknown coordinate: square, often two valuesVolumesPyramid V=13AhV=\frac13AhV=31AhCone V=13πr2hV=\frac13\pi r^2hV=31πr2hSphere V=43πr3V=\frac43\pi r^3V=34πr3Hemisphere 23πr3\frac23\pi r^332πr3Composite: add the volumesSurface areaCone curved surface πrl\pi rlπrlSlant height l=r2+h2l=\sqrt{r^2+h^2}l=r2+h2Sphere 4πr24\pi r^24πr2; hemisphere curved 2πr22\pi r^22πr2Composite: exposed surfaces only3D geometrysolids and anglesVVVAAATriangles in solidsSL uses right-angled triangles onlySquare base 10: half-diagonal 525\sqrt252Cuboid space diagonal a2+b2+c2\sqrt{a^2+b^2+c^2}a2+b2+c2Name the triangle, e.g. VOAVOAVOALine and planeAngle to plane: line and its projectionDrop a perpendicular to the planeCamera C(0,0,9)C(0,0,9)C(0,0,9), Q(20,12,0)Q(20,12,0)Q(20,12,0): sinθ=925\sin\theta=\frac{9}{25}sinθ=259θ=21.1∘\theta=21.1^\circθ=21.1∘Exam tipsForgetting the 13\frac1331 in cones and pyramidsSphere volume uses r3r^3r3Use slant height in πrl\pi rlπrl, not hhhDo not count joined facesSketch and shade the right-angled triangle