← Topic 4 · Geometry
AMC 10/12 · Cheatsheet

Topic 4 · Geometry

Chapter 4 · Solids & analytic depth

📋 Reference · always available
Volumes
Bh,    13Bh,    43πr3Bh,\;\; \tfrac13 Bh,\;\; \tfrac43\pi r^3
Sphere SA
4πr24\pi r^2
Point–line distance
d=∣ax0+by0+c∣a2+b2d=\tfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}
Line–circle
Substitute → quadratic; discriminant gives 0/1/2 intersections.
Sphere in cone (inradius)
r=RhR+R2+h2r = \tfrac{Rh}{R + \sqrt{R^2+h^2}}
Sphere : cylinder volume
VS:VC=2:3V_S : V_C = 2:3
Frustum volume
V=πh3(R2+Rr+r2)V = \tfrac{\pi h}{3}(R^2 + Rr + r^2)
Parabola
y=x24p:focus (0,p), directrix y=−py = \tfrac{x^2}{4p}: \text{focus }(0,p),\text{ directrix }y{=}{-}p
Ellipse
x2a2+y2b2=1,  c2=a2−b2\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1,\;c^2 = a^2 - b^2
Hyperbola
x2a2−y2b2=1,  c2=a2+b2\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1,\;c^2 = a^2 + b^2
Eccentricity
e=c/a:  parabola =1,  ellipse <1,  hyperbola >1e = c/a:\;\text{parabola }=1,\;\text{ellipse }<1,\;\text{hyperbola }>1