IB›IB Maths: Analysis and Approaches SL›Mind maps3.1 Three-dimensional geometryIB Maths: Analysis and Approaches SL: 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 = \mathbf{14}AB=14Unknown coordinate: square both sides, often two valuesVolumesPyramid: V=13AhV = \frac{1}{3}AhV=31AhCone: V=13πr2hV = \frac{1}{3}\pi r^2 hV=31πr2hSphere: V=43πr3V = \frac{4}{3}\pi r^3V=34πr3Hemisphere: 23πr3\frac{2}{3}\pi r^332πr3Composite solid: add the partsSurface areaCone curved surface: πrl\pi r lπrlSlant height l=r2+h2l = \sqrt{r^2 + h^2}l=r2+h2Sphere: 4πr24\pi r^24πr2; hemisphere curved: 2πr22\pi r^22πr2Composite: count exposed surfaces only3D geometrysolids and spaceVVVdddTriangles in solidsSL: right-angled triangles onlySquare base side 10: half-diagonal 525\sqrt{2}52Cuboid space diagonal: a2+b2+c2\sqrt{a^2 + b^2 + c^2}a2+b2+c2Name each triangle by its verticesLine and planeAngle = line and its projection on the planeDrop a perpendicular to the planeJoin its foot to where the line meets the planeUse the right-angled triangle formedExam tipsDo not drop the 13\frac{1}{3}31 in pyramid and coneSphere volume uses r3r^3r3Use slant height, not hhh, in πrl\pi r lπrlHidden joined faces are not countedSketch and shade the triangle used