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geometry

Coordinate geometry, areas & volumes of standard figures and solids.

coordinate geometry · plane figures · solids

Coordinate geometry

Distance & midpoint. 2D/3D distance \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\); midpoint \(M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)\).

Circle. \((x-a)^2+(y-b)^2=r^2\): centre \((a,b)\), radius \(r\).

Pythagoras. \(a^2+b^2=c^2\).

Gradient angle. \(m=\tan\theta\) — the angle a line of gradient \(m\) makes with the positive \(x\)-axis.

Plane figures

Triangle. \(A=\frac12 bh=\frac12 ab\sin C\).

Parallelogram & trapezium. Parallelogram \(A=bh\); trapezium \(A=\frac12(a+b)h\).

Circle. \(A=\pi r^2\), circumference \(2\pi r\).

Arc & sector. Arc length \(s=r\theta\), sector area \(A=\frac12 r^2\theta\), for \(\theta\) in radians.

Solids

Prism & cylinder. \(V=Ah\); cylinder \(V=\pi r^2h\), curved surface area \(2\pi rh\).

Pyramid & cone. \(V=\frac13 Ah\); cone \(V=\frac13\pi r^2h\), curved surface area \(\pi rl\), slant height \(l^2=r^2+h^2\).

Sphere & hemisphere. Sphere \(V=\frac43\pi r^3\), surface area \(4\pi r^2\); hemisphere \(V=\frac23\pi r^3\), total surface area \(3\pi r^2\).