Mensuration (2D)
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Areas of triangles
General triangle
K=\frac12 b h
Equilateral triangle
K=\frac{\sqrt3}{4}a^2,\quad h=\frac{\sqrt3}{2}a
Heron's formula
s=\frac{a+b+c}{2},\quad K=\sqrt{s(s-a)(s-b)(s-c)}
Radii of an equilateral triangle
r_{\text{in}}=\frac{a}{2\sqrt3},\quad R=\frac{a}{\sqrt3}
Areas of quadrilaterals
Rectangle
K=lb,\quad P=2(l+b),\quad d=\sqrt{l^2+b^2}
Square
K=a^2=\frac{d^2}{2},\quad d=a\sqrt2
Rhombus
K=\frac{d_1d_2}{2}
Trapezium
K=\frac{(a+b)}{2}h
Cyclic quadrilateral (Brahmagupta)
K=\sqrt{(s-a)(s-b)(s-c)(s-d)}
Circles, sectors and rings
Circle
K=\pi r^2,\quad C=2\pi r=\pi d
Sector
K=\frac{\theta}{360}\pi r^2,\quad \ell=\frac{\theta}{360}\,2\pi r
Segment
K=\frac{\theta}{360}\pi r^2-\frac12 r^2\sin\theta
Ring / path
K=\pi(R^2-r^2)=\pi(R+r)(R-r)
Rolling wheel
\text{distance}=n\pi d
Regular polygons and inscribed figures
Regular hexagon
K=\frac{3\sqrt3}{2}a^2=6\times\frac{\sqrt3}{4}a^2
Regular octagon
K=2(1+\sqrt2)a^2
Square in a circle
\text{diagonal}=\text{diameter}=2R
Circle in a square
d_{\text{circle}}=a
Circle in an equilateral triangle
r=\frac{a}{2\sqrt3}
Percentage change, similarity and re-bent shapes
Scaling
\ell\to k\ell\ \Rightarrow\ K\to k^2K,\quad P\to kP
Successive change in area
\Delta\%=a+b+\frac{ab}{100}
Equal perimeter
P_1=P_2\ \text{(re-bent wire)}
Side change from area change
\Delta a\%=\left(\sqrt{1+\frac{\Delta K}{100}}-1\right)\times100