Geometry and mensuration

Any simple polygon with $n$ sides (sides do not cross), convex or not: $180^{\circ }$ for a triangle, $360^{\circ }$ for a quadrilateral, $540^{\circ }$ for a pentagon.

One exterior angle taken at each vertex of a convex polygon; the sum is $360^{\circ }$ whatever the value of $n$. At every vertex, interior angle $+$ exterior angle $=180^{\circ }$.

Regular polygon only (all sides equal and all angles equal).

A diagonal joins two vertices that are not neighbours. Each vertex joins $n-3$ others by a diagonal (not itself and not its two neighbours); dividing by 2 stops each diagonal being counted twice.

$c$ is the side opposite angle $C$. Apply the test to the longest side: the triangle is acute, right or obtuse according to the angle opposite that side.

$h$ is the perpendicular height onto the base $b$ (in an obtuse triangle it can fall outside the triangle, on the extended base); $C$ is the angle between sides $a$ and $b$.

Any triangle whose three sides are known. If any of $s-a$, $s-b$, $s-c$ is zero or negative, the three lengths do not form a triangle.

$h$ is the height onto the base $b$; it bisects both the base and the angle between the equal sides. The lengths form a triangle only when $b<2a$.

All four centres coincide at the centroid, which divides each height in the ratio $2:1$ from the vertex. So $R=\frac{2}{3}h$ and $r=\frac{1}{3}h$, giving $R=2r$ and $h=3r=\frac{3}{2}R$.

Any triangle: $r$ is the radius of the incircle (touches all three sides), $R$ the radius of the circumcircle (passes through all three vertices) and $s$ the semi-perimeter.

$h_c$ is the altitude from the right angle to the hypotenuse. If it cuts the hypotenuse into parts $p$ (next to leg $a$) and $q$ (next to leg $b$), then $h_c^2=pq$, $a^2=pc$ and $b^2=qc$.

Any triangle. The cosine rule becomes Pythagoras' theorem when $C=90^{\circ }$.

$D$ is the mid-point of $BC$, so $BD=a/2$. In a right triangle the median to the hypotenuse is half the hypotenuse.

$AD$ is the internal bisector of angle $A$, meeting $BC$ at $D$: it divides the opposite side in the ratio of the two sides that form the angle, and the second relation gives its length. When $AB\ne AC$, the external bisector of angle $A$ meets $BC$ produced at a point $E$ with $BE/EC=AB/AC$.

Parallelogram: $\theta$ is the angle between sides $a$ and $b$, and $h$ the height onto base $b$. Trapezium: $a$ and $b$ are the parallel sides and $h$ the distance between them. Any quadrilateral: $\theta$ is the angle between the diagonals ($90^{\circ }$ for a rhombus, a square or a kite).

$d$ is the diagonal. For the ring, $R$ and $r$ are the outer and inner radii.

Only for a cyclic quadrilateral (opposite angles supplementary), with sides $a,b,c,d$. Putting $d=0$ gives Heron's formula.

Regular polygons only. $n=4$ gives $a^2$ and $n=6$ gives $\frac{3\sqrt{3}}{2}{a}^2$.

A path of width $w$ running all round the outside or the inside of a rectangular plot $l\times b$. The crossing roads are two roads of width $w$ through the plot, one parallel to each pair of sides; the square where they cross is counted once, which is why $w^2$ is subtracted. A circular path of width $w$ round the outside of a circle of radius $r$ has area $\pi \left[(r+w{)}^2-r^2\right]=\pi w(2r+w)$.

Radius $r$, diameter $d$. A wheel of radius $r$ rolling without slipping moves forward $2\pi r$ in one complete revolution.

$\theta$ is the angle at the centre, in degrees. With $\theta$ in radians the same results read $l=r\theta$ and $A=\frac{1}{2}{r}^2\theta$.

Minor segment cut off by a chord that subtends $\theta$ (degrees) at the centre: sector minus triangle. $d$ is the perpendicular distance of the chord from the centre. Major segment $=\pi r^2-$ minor segment.

$PT$: tangent from a point at distance $d$ from the centre of a circle of radius $r$. For two circles of radii $r_1,r_2$ with centres $d$ apart: a direct common tangent has both circles on the same side of it and exists only if $d\ge |r_1-r_2|$; a transverse common tangent passes between them and exists only if $d\ge r_1+r_2$. Each length is measured between the two points of contact.

Between $A(x_1,y_1)$ and $B(x_2,y_2)$. The distance of $(x,y)$ from the origin is $\sqrt{x^2+y^2}$.

$P$ divides $AB$ in the ratio $m:n$, that is $AP:PB=m:n$, from $A(x_1,y_1)$ towards $B(x_2,y_2)$. External division needs $m\ne n$. The ratio $1:1$ gives the mid-point.

Triangle $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ with $a=BC$, $b=CA$, $c=AB$: in the incentre, each vertex is weighted by the length of the side opposite it.

Take the absolute value: the expression inside is negative when the vertices are listed clockwise. $\Delta =0$ means the three points are collinear. For a polygon, list the vertices in order round the boundary and use the shoelace sum $\frac{1}{2}\left|\sum (x_i{y}_{i+1}-x_{i+1}{y}_i)\right|$, returning to the first vertex at the end.

$\theta$ is the angle the line makes with the positive $x$-axis, measured anticlockwise. A horizontal line has $m=0$; a vertical line has no defined slope. The perpendicularity condition applies when neither line is vertical (a horizontal line is perpendicular to a vertical one).

The two-point form needs $x_1\ne x_2$ (otherwise the line is $x=x_1$). The intercept form needs $a\ne 0$ and $b\ne 0$, so it cannot describe a line through the origin or a line parallel to an axis. A horizontal line is $y=k$ and a vertical line is $x=k$. The line $\frac{x}{a}+\frac{y}{b}=1$ forms a right triangle of area $\frac{1}{2}|ab|$ with the coordinate axes.

Perpendicular distance of $(x_1,y_1)$ from $Ax+By+C=0$. For the parallel lines $Ax+By+C_1=0$ and $Ax+By+C_2=0$, first make the $x$ and $y$ coefficients identical in both equations.

$\varphi$ is the acute angle between lines of slopes $m_1$ and $m_2$. If $1+m_1{m}_2=0$ the lines are perpendicular.

Centre $(h,k)$, radius $r$. The general form is a real circle only when $g^2+f^2-c>0$ (it is a single point when this is zero). A point lies inside, on or outside the circle according as its distance from the centre is less than, equal to or greater than $r$.

The diagonal here is the space (body) diagonal, from a corner to the opposite corner through the interior; a face diagonal of a cube is $a\sqrt{2}$. The four walls of a room of length $l$, breadth $b$ and height $h$ have area $2h(l+b)$.

Hollow cylinder (pipe) of outer radius $R$, inner radius $r$ and length $h$: volume of material $=\pi h(R^2-r^2)$.

$h$ is the vertical height and $l$ the slant height; do not interchange them. A cone has one third of the volume of a cylinder with the same base and height.

A solid hemisphere's total surface includes its flat circular face ($\pi r^2$). Spherical shell with outer radius $R$ and inner radius $r$: volume of material $=\frac{4}{3}\pi (R^3-r^3)$.

The part of a cone between its base and a cut parallel to the base; $R$ and $r$ are the radii of the two circular ends and $h$ is the perpendicular distance between them. Putting $r=0$ gives the cone formulas.

$h$ is the perpendicular height; both volume formulas hold for any base, and for a slanting (oblique) solid too. Lateral surface of a right prism $=$ base perimeter $\times h$; of a regular pyramid $=\frac{1}{2}\times$ base perimeter $\times$ slant height, where the slant height is the height of each triangular face, not the length of a sloping edge.

Any convex polyhedron with $V$ vertices, $E$ edges and $F$ faces. Cube: $8-12+6=2$; triangular prism: $6-9+5=2$; square-based pyramid: $5-8+5=2$.

Holds only when ALL linear dimensions change by the same factor. Doubling the radius of a sphere multiplies its surface area by 4 and its volume by 8.

Every length changes by $x\%$ of its original value ($x<0$ for a decrease). A $10\%$ increase in every length raises area by $21\%$ and volume by $33.1\%$; a $10\%$ decrease lowers area by $19\%$ and volume by $27.1\%$.