Quadrilaterals

1. The parallelogram

$ABCD\,\left( AB||\,CD,BC||\,AD,\,DE\bot AB \right);$

$\begin{aligned} & AC\cap BD=\left\{ O \right\},\, \\ & OA=OC,\,\,OB=OD\,;\,{{A}_{ABCD}}=AB\cdot DE;\, \\ & {{A}_{ABCD}}=AB\cdot AD\cdot \sin A; \\ \end{aligned}$

2. The rectangle

$ABCD\,\left( AB||CD,BC||AD,\measuredangle \,A={{90}^{0}} \right)$;

$\begin{aligned} & AC=BD; \\ & {{A}_{ABCD}}=AB\cdot AD; \\ \end{aligned}$

3. The rhombus $ABCD\,\left( AB||CD,\,BC||AD,\,AB=BC \right)\,:\,AC={{d}_{1}},$

$\begin{aligned} & BD={{d}_{2}},\,AB=AD=a;\,AC\bot BD; \\ & {{A}_{ABCD}}=\frac{{{d}_{1}}\cdot {{d}_{2}}}{2}\,; \\ \end{aligned}$

4. The square $ABCD\,(AB||CD,BC||AD,AB=BC,\measuredangle \,A={{90}^{0}},AB=a,AC=d)$

$AC=BD,\text{ }AC\bot BD,\text{ }d=a\sqrt{2};\text{ }{{A}_{ABCD}}={{a}^{2}};$

5. The trapezium

$ABCD\left( AB||CD,AB=B,DC=b,DE=h,MN-linie\,\,\,\,mijlocie \right):$

$MN=\frac{B+b}{2};\,\text{ }{{A}_{ABCD}}=\frac{\left( B+b \right)\cdot h}{2}=MN\cdot h;$

Polygons inscribed in a circle

1. The cyclic quadrilateral

$\begin{aligned} & \measuredangle \,BAD+\measuredangle \,BCD={{180}^{0}}; \\ & \measuredangle \,BAC\,\equiv \,\measuredangle \,BDC; \\ \end{aligned}$

Ptolemy's theorem:

$AB\cdot DC+AD\cdot BC=AC\cdot BD;$

${{A}_{ABCD}}=\frac{1}{2}AC\cdot BD\cdot \sin \,\alpha$

2. Regular polygons inscribed in a circle of radius R

The equilateral triangle: ${{l}_{3}}=R\sqrt{3},{{a}_{3}}=\frac{R}{2},{{S}_{3}}=\frac{3\sqrt{3}{{R}^{2}}}{4};$

The square: ${{l}_{4}}=R\sqrt{2}, {{a}_{4}}=\frac{R\sqrt{2}}{2}, {{S}_{4}}=2{{R}^{2}};$

The regular hexagon: ${{l}_{6}}=R,{{a}_{6}}=\frac{R\sqrt{3}}{2},{{S}_{6}}=\frac{3\sqrt{3}{{R}^{2}}}{2};$

The regular polygon with n sides: ${{l}_{n}}=2R\sin \frac{\pi }{n},{{a}_{n}}=R\cos \frac{\pi }{n};$

$S=\frac{n}{2}{{R}^{2}}\sin \frac{2\pi }{n}=p\cdot {{a}_{n}}$, where $p=\frac{n\cdot {{l}_{n}}}{2}$

The circle

Lengths and areas: ${{l}_{cerc}}=2\pi R;\text{ }{{A}_{cerc}}=\pi {{R}^{2}};$

${{l}_{arcAB}}=\frac{\pi R\alpha }{180},$

$\alpha$ – measure in degrees;

${{A}_{\sec torAB}}=\frac{\pi {{R}^{2}}\alpha }{360};$

$\mu \left( \measuredangle AOB \right)=\frac{\alpha \cdot \pi }{180}$ ($\mu$ – measure in radians);

Angle with vertex inside the circle:

$m\left( \measuredangle AOB \right)=m\left( \overset\frown{AB} \right):$

$m\left( \measuredangle AMB \right)=\frac{m\left( \overset\frown{AB} \right)+m\left( \overset\frown{CD} \right)}{2};$

Angle with vertex on the circle $\left( OM\bot MT \right);$

$m\left( \measuredangle AOB \right)=\frac{m\left( \overset\frown{AB} \right)}{2};$

$m\left( \measuredangle AMT \right)=\frac{m\left( \overset\frown{AM} \right)}{2};$

Angle with vertex outside the circle $\left( OT\bot MT \right):$

$m\left( \measuredangle AMB \right)=\frac{m\left( \overset\frown{AM} \right)-m\left( \overset\frown{CD} \right)}{2};$

$m\left( \measuredangle BMT \right)=\frac{m\left( \overset\frown{BT} \right)-m\left( \overset\frown{DT} \right)}{2};$

The power of a point with respect to a circle $\left( OT\bot MT \right):$

$p\left( M \right)=MA\cdot MB=O{{M}^{2}}-{{r}^{2}}=M{{T}^{2}};$

$P\left( N \right)=NA\cdot NB={{r}^{2}}-O{{N}^{2}};$

Circular sector

A circular sector is a portion of a disc of angle $\theta <\pi$, as shown in the adjacent figure.

$R=$ the radius of the circle, $s=$ the length of the arc, $h=$ the height of the circular segment, $d=$ the height of the triangular portion, $A=$ the area of the sector; then:

$R=h+d$ $s=R\cdot \theta$

$d=R\cdot \cos \left( \frac{\theta }{2} \right)=\frac{1}{2}\sqrt{4{{R}^{2}}-{{c}^{2}}}$ $c=R\cdot \sin \left( \frac{\theta }{2} \right)=2\sqrt{h(2R-h)}$

$\theta =\frac{s}{R}=2\arccos \left( \frac{d}{R} \right)$ $A=\frac{1}{2}R\cdot s=\frac{1}{2}{{R}^{2}}\cdot \theta$

Further plane geometry

The orthic triangle is the triangle determined by the feet of the altitudes of a triangle; of all triangles with their vertices on the sides of a triangle (or on their extensions), the orthic triangle has the smallest perimeter;

A cevian is the line determined by a vertex of a triangle and a point of the opposite side;

Ceva's theorem: The cevians AM, BN, CP of the triangle ABC are concurrent if and only if $\frac{MB}{MC}\cdot \frac{NC}{NA}\cdot \frac{PA}{PB}=1;$

Menelaus' theorem: On the lines BC, CA, AB determined by the sides of the triangle ABC, consider the points M, N and P respectively, two of them on the sides of the triangle and one on the extension of a side, or all three on extensions of the sides. The points M, N, P are collinear if and only if:

$\frac{MB}{MC}\cdot \frac{NC}{NA}\cdot \frac{PA}{PB}=1;$

Euler's line: In any triangle, the points H, O and G (the orthocentre, the circumcentre and the centroid) are collinear;

The Simson line: The projections of a point on the circumcircle of a triangle onto the lines containing its sides are collinear;

The excircle of a triangle is tangent to one side of the triangle and to the extensions of the other two sides; the centre of the excircle is the intersection of the bisector of one interior angle with the bisectors of the other two exterior angles;

The Euler circle (the nine-point circle): the feet of the altitudes of a triangle, the midpoints of the sides, and the midpoints of the segments determined by the orthocentre and the vertices are concyclic;

Polyhedra

I The prism

1. The rectangular cuboid:

${{A}_{lat}}=2\left( a+b \right)\cdot c;$

${{A}_{tot}}=2\left( ab+ac+bc \right);$

$V=abc;$

${{d}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}};$

2. The cube (of edge $a=b=c$):

$A=6{{a}^{2}};$

$V={{a}^{3}};$

$d=a\sqrt{3;}$

3. The parallelepiped:

$BO\bot \left( ABC \right),B'O=h;$

$V={{A}_{ABCD}}\cdot h;$

4. The prism (right or oblique, of height h):

$V={{A}_{bazei}}\cdot h;$

5. The regular triangular prism $\left( AB=a,\Delta ABC=echilateral \right):$

${{A}_{lat}}=3a\cdot h\,;{{A}_{tot}}=3a\cdot h+\frac{{{a}^{2}}\sqrt{3}}{2};$

$V=\frac{{{a}^{2}}\sqrt{3}}{4}\cdot h\,;$

II The pyramid

1. The regular tetrahedron (all edges congruent, $AO\bot \left( BCD \right)$, $AM\bot DC$):

$h=\frac{a\sqrt{6}}{3},AM=\frac{a\sqrt{3}}{2};$

$\sin A\hat{B}O=\frac{\sqrt{6}}{3},$

$\sin A\hat{M}O=\frac{2\sqrt{2}}{3};$

$A={{a}^{2}}\sqrt{3};V=\frac{{{a}^{3}}\sqrt{2}}{12};$

2. The trirectangular tetrahedron ($OA\bot OB\bot OC\bot OA,$ $OA=OB=OC=a$, $CM\bot AB$):

ΔABC = equilateral,

$OM=\frac{a\sqrt{2}}{2},CM=\frac{a\sqrt{6}}{2},$

$AB=a\sqrt{2};{{A}_{ABC}}=\frac{{{a}^{2}}\sqrt{3}}{2};$

$Atot=\frac{3{{a}^{2}}}{2}+\frac{{{a}^{2}}\sqrt{3}}{2};$

$V=\frac{{{a}^{3}}}{6}$;

3. The regular triangular pyramid ($AB=AC=BC=a$, $VA=VB=VC$, $VM\bot BC$, $VM-apotema$):

$VM=\sqrt{{{h}^{2}}+\frac{{{a}^{2}}}{12}};$

${{A}_{lat}}=\frac{3a\cdot VM}{2};$

${{A}_{tot}}=\frac{{{a}^{2}}\sqrt{3}}{4}+\frac{3a\cdot VM}{2};$

$V=\frac{{{a}^{2}}\sqrt{3}}{4}\cdot \frac{h}{3};$

4. The regular square pyramid ($ABCD$ – a square of side a, $VA=VB=VC=VD$, $VM\bot BC$):

${{A}_{lat}}=2a\cdot VM;$

${{A}_{tot}}={{a}^{2}}+2a\cdot VM;$

$V=\frac{{{a}^{2}}\cdot h}{3};$

5. The regular hexagonal pyramid ($ABCDEF$ – a regular hexagon, $VA=VB=VC=VD=VE=VF=a$, $VM\bot BC$):

$VM=\sqrt{{{h}^{2}}+\frac{3{{a}^{2}}}{4}};$

${{A}_{tot}}=3a\cdot VM;$

${{A}_{tot}}=\frac{3{{a}^{2}}\sqrt{3}}{2}+3a\cdot VM;$

$V=\frac{{{a}^{2}}\sqrt{3}\cdot h}{2};$

6. The regular pyramid (the foot of the altitude coincides with the circumcentre of the base):

${{A}_{lat}}=\frac{{{P}_{bazei}}\cdot apotema}{2};$

${{A}_{tot}}={{A}_{bazei}}+{{A}_{lat}}\,;\,V=\frac{{{A}_{bazei}}\cdot h}{3};$

7. The pyramid (of height h)

${{A}_{tot}}={{A}_{bazei}}+{{A}_{lat}}\,;\,V=\frac{{{A}_{bazei}}\cdot h}{3};$

III The frustum of a pyramid (B – area of the larger base, b – area of the smaller base,

h – height):

1. The frustum of a general pyramid:

$V=\frac{h}{3}\left( B+b+\sqrt{B\cdot b} \right)$

2. The frustum of a regular pyramid (P – perimeter of the larger base, p – perimeter of the smaller base, a_p – slant height):

${{A}_{lat}}=\frac{\left( P+p \right)\cdot {{a}_{p}}}{2};$

${{A}_{tot}}=B+b+\frac{\left( P+p \right)\cdot {{a}_{p}}}{2};$

$V=\frac{h}{3}\left( B+b+\sqrt{B\cdot b} \right);$

IV Regular polyhedra

Euler's relation: $v-m+f=2$

(v – number of vertices, m – number of edges, f – number of faces);

The types of regular polyhedra:

• the regular tetrahedron: $f=4,v=4,m=6;$

• the cube (regular hexahedron): $f=6,v=8,m=12;$

• the regular octahedron: $f=8,v=6,m=12;$

• the regular dodecahedron: $f=12,v=20,m=30;$

• the regular icosahedron: $f=20,v=12,m=30;$

Solids of revolution

Notation: R – radius, G – slant height, h – height

1) The right circular cylinder ($h=G$):

${{A}_{lat}}=2\pi R\cdot G;$

${{A}_{tot}}=2\pi R\cdot \left( R+G \right);$

$V=\pi {{R}^{2}}\cdot h;$

2) The right circular cone

${{G}^{2}}={{h}^{2}}+{{R}^{2}};{{A}_{lat}}=\pi R\cdot G;$

${{A}_{tot}}=\pi R\cdot \left( R+G \right);$

$V=\frac{\pi {{R}^{2}}\cdot h}{3};$

3) The frustum of a cone (r – radius of the smaller base):

${{G}^{2}}={{h}^{2}}+{{\left( R-r \right)}^{2}};$

${{A}_{lat}}=\pi G\cdot \left( R+r \right);$

${{A}_{tot}}=\pi G\cdot \left( R+r \right)+\pi \left( {{R}^{2}}+{{r}^{2}} \right);$

$V=\frac{\pi h}{3}\left( {{R}^{2}}+{{r}^{2}}+R\cdot r \right);$

4) The sphere:

$A=4\pi {{R}^{2}};V=\frac{4\pi {{R}^{3}}}{3};$

${{A}_{calotei\,\,sferei}}=2\pi R{{h}_{1}};$

${{A}_{zonei}}=2\pi R{{h}_{2}};$