Toolkit 823D Shapes: Surface Area and VolumeTotal Surface Area (TSA) and VolumeCubeTSA=6a2\mathrm{TSA}=6a^2TSA=6a2V=a3V=a^3V=a3Rectangular Prism (Cuboid)TSA=2(ab+ac+bc)\mathrm{TSA}=2(ab+ac+bc)TSA=2(ab+ac+bc)V=abcV=abcV=abcPrismV=h⋅AreabaseV=h\cdot \text{Area}_{\text{base}}V=h⋅AreabaseSphereTSA=4πr2\mathrm{TSA}=4\pi r^2TSA=4πr2V=43πr3V=\frac{4}{3}\pi r^3V=34πr3CylinderTSA=2πr(r+h)\mathrm{TSA}=2\pi r(r+h)TSA=2πr(r+h)Lateral Area=2πrh\text{Lateral Area}=2\pi rhLateral Area=2πrhV=πr2hV=\pi r^2hV=πr2hConeTSA=πr(r+l)\mathrm{TSA}=\pi r(r+l)TSA=πr(r+l)Lateral Area=πrl\text{Lateral Area}=\pi rlLateral Area=πrll=r2+h2l=\sqrt{r^2+h^2}l=r2+h2V=13πr2hV=\frac{1}{3}\pi r^2hV=31πr2hPyramidV=13Areabase⋅hV=\frac{1}{3}\text{Area}_{\text{base}}\cdot hV=31Areabase⋅hRegular TetrahedronTSA=a23\mathrm{TSA}=a^2\sqrt{3}TSA=a23h=a23h=a\sqrt{\dfrac{2}{3}}h=a32V=a3212V=\frac{a^3\sqrt{2}}{12}V=12a32Triangular PrismV=12Areabase×LV=\frac12\text{Area}_{\text{base}}\times LV=21Areabase×LRelated ProblemsCoreMATHCOUNTS State Team 2025 (Problem 9)MajorAMC 12A 2025 (Problem 20)AMC 10B/12B 2025 (Problem 19/15)
