Quadrilaterals

1. The parallelogram

2. The rectangle

;

3. The rhombus

4. The square

5. The trapezium

Polygons inscribed in a circle

1. The cyclic quadrilateral

Ptolemy's theorem:

2. Regular polygons inscribed in a circle of radius R

The equilateral triangle:

The square:

The regular hexagon:

The regular polygon with n sides:

, where

The circle

Lengths and areas:

– measure in degrees;

( – measure in radians);

Angle with vertex inside the circle:

Angle with vertex on the circle

Angle with vertex outside the circle

The power of a point with respect to a circle

Circular sector

A circular sector is a portion of a disc of angle , as shown in the adjacent figure.

the radius of the circle, the length of the arc, the height of the circular segment, the height of the triangular portion, the area of the sector; then:

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

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:

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:

2. The cube (of edge ):

3. The parallelepiped:

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

5. The regular triangular prism

II The pyramid

1. The regular tetrahedron (all edges congruent, , ):

2. The trirectangular tetrahedron ( , ):

ΔABC = equilateral,

;

3. The regular triangular pyramid (, , , ):

4. The regular square pyramid ( – a square of side a, , ):

5. The regular hexagonal pyramid ( – a regular hexagon, , ):

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

7. The pyramid (of height h)

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:

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

IV Regular polyhedra

Euler's relation:

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

The types of regular polyhedra:

• the regular tetrahedron:

• the cube (regular hexahedron):

• the regular octahedron:

• the regular dodecahedron:

• the regular icosahedron:

Solids of revolution

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

1) The right circular cylinder ():

2) The right circular cone

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

4) The sphere: